The article, by Patrick M. O'Neil, was published in The Journal of Libertarian Studies, Spring 1983, pp. 81-99. A full, scholarly analysis of that article would require cognizance of later works by other leading Objectivists, such as Leonard Peikoff's OPAR and books by Tara Smith and Craig Biddle. What I can do here, however, is indicate in general terms the most fundamental reason why the article does not comprehend the objectivity of the Objectivist ethics.
The article states (p. 81);
In this essay, the claims of Objectivism to present an objective epistemology will not he considered, but only the claim of Rand's ethical system to represent an objective value system.
As a comment in this thread has already pointed out, the claim of Objectivism to objectivity in ethics cannot be understood and validated without an understanding of Objectivist epistemology, specially, Ayn Rand's theory of concepts and how that theory applies to the most fundamental concept in ethics, the concept of 'value.'
Jumping ahead to p. 87, we find what I regard as the most fundamental issue on which the article questions Objectivism's objectivity:
The whole Randian moral system rests upon the most basic moral command that one ought to do that which preserves one's life (qua man). It makes no difference, then, that the standard is not variable in response to the individual, subjective will, for the commitment to the command ("one ought to behave so as to survive qua man") demands a more basic deontological moral imperative -- setting up the prospect of an infinite regression of moral commands -- for a rational creature has the option to choose nonsurvival as well as survival: "Metaphysically, the choice 'to be conscious or not' is the choice of life or death." 
Objectivism identifies life as the "ultimate value," an end in itself requiring no further end to "justify" it. The O'Neil article does not comprehend how that can be logically possible, since man surely has the capacity to choose death rather than life.
Objectivism can reply: what code of values logically follows from the choice to die? The answer is none. There is nothing man must do to die. He needs only to refrain from acting at all (insofar as his actions are open to his choice, as most of them are). Remember that the whole Objectivist ethics rests on the question: why does man need values? In VOS Chap. 1, Ayn Rand states:
The first question that has to be answered, as a precondition of any attempt to define, to judge or to accept any specific system of ethics, is: Why does man need a code of values?
The O'Neil article seems to ask: why, then, can it be said that man ought to seek to sustain his life for as long as he can? But this misses the point that Objectivism identifies. There is no objective, reality-based foundation for any "ought" of any kind except in the context of seeking to sustain one's life.
Objectivism develops this point further in the analysis of the concept 'value' and what it contextually depends on.
If there is no objective moral basis for choosing to live, then it may seem that there can likewise be no objective moral basis for morally condemning a choice not to live. OPAR discusses this point further in Chapter 7. "The Good," in the section titled, "Value as Objective." specifically on pp. 247-248. Any objective moral condemnation of any actions performed by a "death-chooser" must necessarily be reached from the context of life-seekers who hold life as their standard of value, and the condemnation would generally be limited to any anti-life actions which a death-chooser may perform that endanger the lives and well being of life-seekers. The latter are perfectly morally entitled to defend themselves from death-choosers in any manner that might be objectively needed for self-preservation, just as life-seekers would be morally entitled to do so against attacks by wild animals. When a death-chooser seeks to attack life-seekers (which he has absolutely no logical need or reason to do, but no reason to remain logical, either), he is acting as a wild animal (actually worse, since a wild animal acts on the automatic goal of life-seeking). OPAR describes a death-chooser as monstrous (by the moral standards of life-seekers), and that is an objective moral judgment based on life as the only objectively valid standard of value.
The analysis by O'Neil might go on to claim that a death-chooser is morally free to do whatever he wants because no moral code follows from death-choosing, and that he is being denied his moral freedom and right to act in accord with his own standard when he attacks life-seekers and they defend themselves. But what is a death-chooser? He is one who does not seek to live. He therefore has no moral reason to act at all, no need to act to gain and/or keep anything, and a stated wish to die. If he wants to die by suicide, he is free to do so. If he wants "death by the police" because he is initiating physical force against others and can't be stopped any other way, let his wish be granted. That is the moral evaluation that applies to death-choosers. (I also hasten to add that a death-chooser who was actually a life-seeker throughout his life up to a certain point, and who may yet choose to reverse his course, may warrant life-seeking assistance from any life-seekers who care about him and want to help.)
In short, I maintain that in order to understand an objective view of death-choosers, one must first understand an objective view of wild animals who attack humans. Death-choosers start from there and descend at least one "rung" lower on the moral scale, since they have the capacity to choose to live and to act rationally in pursuit of life.
(It should also be noted that the whole idea of a "right" to life, including the right to liberty and the pursuit of happiness, proceeds from the context of morality for life-seekers. Death-choosers have no such "right," nor any need of it, except insofar as they retain the capacity to choose life.)
In a comment, the questioner asks about the allegedly "objectivist assertion" that "we ought to value life."
There is a factual error in this comment-question: Objectivism doesn't exactly say that -- and a logical problem: the conclusion doesn't seem to follow from the premises as stated. The alleged argument seems to be as follows:
(a) Life is a necessary condition for values.
But why does either conclusion, valuing or disvaluing, follow logically from the "necessity" (or precondition) observation? At most, all we can say from the argument is that if we want to have values, then we must value life; and, likewise, if we want to have disvalues, then we must again value life, since life is claimed to be necessary for disvalues as well as for values. We still have a choice about whether to have values (or disvalues) or not. How do (a), (b) and/or (c) necessarily prove anything about how we "ought" to choose? If we seek neither values nor disvalues, then why would we need to value life?
This, in turn, leads directly to the crux of the Objectivist view. In the words of John Galt (in Atlas Shrugged): "My morality, the morality of reason, is contained in a single axiom: existence exists -- and in a single choice: to live. The rest proceeds from these." Objectivism does not say that man "ought" to value life. It says that man has a choice to live or not, and that all his other values, if they are objective, proceed from the choice to live.
The O'Neil article and similar discussions may ask what makes the initial choice "objective," if there can be no antecedent objective moral principle mandating it. Why is the choice to live objectively valid? Wouldn't the choice not to live be just as valid and objective? The Objectivist answer, as I understand it, is to comprehend the objective meaning of the choice. The choice not to live is essentially a choice to go out of existence, to remove oneself from the whole realm of objectivity and validity (and reality). It is a negation, not an affirmation. It is anti-objectivity, anti-validity and anti-reality as well as anti-life. In contrast, the choice to live is an affirmation of reality, objectivity, validity, logic, reason, and all the other essential values and virtues for man's life qua man. Hence, it is not valid to hold the choice to live as objectively residing on the same level as the choice not to live.
This does not mean, however, that there is any moral obligation or mandate to choose to live. The choice to live or not rests entirely with every individual. Those who choose to live will be acting objectively in accord with their nature; those who choose not to live will be acting against it, and incurring consequences of the most severe kind (leaving existence) as a result.
Bear in mind, also, that any code of morality, rational or not, faces the same issue: man always has a choice about whether or not to follow the code, whether or not to be moral. Objectivism grounds the fundamental choice in the objective phenomenon of life, rather than in some non-objective moral dogma.
Furthermore, note that choosing not to live does not give a death-chooser a moral license to do whatever he pleases, if that is taken to mean attacking life-seekers. Life-seekers have a strong interest in moral judgments toward death-choosers insofar as any conflict with death-choosers might arise, although such conflicts are never necessary or inevitable and would tend to imply that the death-chooser isn't acting fully consistently with his stated choice (which wouldn't matter to a death-chooser but most certainly matters to a life-seeker).
In the comments, the questioner asks if Objectivism says that objective moral values can be derived from subjective moral values. By "subjective moral value," the questioner apparently means the choice to live.
To understand where this line of questioning is "coming from," I went back and rechecked the O'Neil paper and its references to Ayn Rand's formulations. In a key TOE formulation quoted in the O'Neil paper (p. 87), Ayn Rand observes:
Epistemologically, the concept of "value" is genetically dependent upon and derived from the antecedent concept of "life." To speak of "value" as apart from "life" is worse than a contradiction in terms. "It is only the concept of 'Life' that makes the concept of 'Value' possible."
The ending sentence in this excerpt comes from Galt's Speech, which continues: "It is only to a living entity that things can be good or evil."
There is, indeed, a presupposition in these formulations that a living entity wants to remain alive. Non-human living entities do so automatically, but man can choose whether or not to act to sustain his life.
Observe what Ayn Rand is actually saying here. She is saying simply that there is a factual relationship between living and valuing, such that the concept of a "value" itself, as well as the entire morality of reason that proceeds from it, arise from that relationship.
Is there any such factual relationship between dying and valuing? The above formulations say there certainly is not. Valuing depends on living; it does not follow from dying. Thus, only one objective moral code is possible: the one that proceeds from living -- even though there are two possible alternatives for man to choose from in the life-death alternative. No moral code of any kind proceeds from dying ("dying" here is meant to refer to a hypothetical human death-chooser, "hypothetical" because it's mostly a rather fantastic fiction in actuality).
To put this more simply, the standard of value is objective, but the choice of an individual human to adhere to it or not (and thereby to live or not) is his own individual choice to make. The objective morality of reason offers no guidance on whether or not an individual should choose to live; it's up to him. But why would anyone need any guidance in such a fundamental, ever-present issue as life or death -- an issue he has faced constantly ever since he was born and had to have chosen already in order to have lived long enough to be able to ask questions about moral codes?
Note also that "man's life qua man as the standard of value for man" does not proceed from any particular individual's choice to live or not, but from his capacity to make that choice, and the implications of the "life" choice (and lack of such implications for the "death" choice).