The practice of meditation breaks the cognitive illusion that most of us live with, i.e,. that we are egos, separate selves, something to which the pronoun "I" can refer to, riding around somewhere inside our bodies. Most people feel like they are located somewhere behind their eyes, looking out into the world that is other than what they are. We know this is an illusion neurologically; we know this is not the case scientifically looking reductively at what the self could possibly be as a collection of systems. Identification with one's thoughts is also the result of this illusion. Could it be that Objectivism -- with its emphasis on heroic individuality -- is based on this illusion?
Objectivism is based most fundamentally on metaphysical axioms -- existence, consciousness and identity -- and reason, the faculty that identifies and integrates the material provided by man's senses. Objectivism does not begin with "self" or "mind" or "ego," although these concepts (which man forms by looking at reality and at himself and conceptualizing what he observes) certainly figure heavily in Objectivism's higher level conclusions in ethics, politics and esthetics.
For a brief summary of Objectivism's view of "Self," refer to that topic in The Ayn Rand Lexicon.
The question also appeals to biological reductionism, which is offered as scientific and thus cannot refute the basic premises of science that make the reductionistic theory possible:
We know this [individual ego or self] is an illusion neurologically; we know this is not the case [by] scientifically looking reductively at what the self could possibly be as a collection of systems.
Identification of "subsystems" that comprise a higher level "system" does not refute the existence of either the composite entity or its constituent entities. Objectivism recognizes no a priori requirement for "systems" to be irreducible. A is A; systems exist and are what they are. Consciousness exists, too (as a faculty of certain living entities), and is what it is.
Update: Primacy of Existence vs. Primacy of Consciousness
In both the original question and in the comments, the questioner has expressed (in unusually stark terms) a perspective known in Objectivism as the "primary of consciousness." Objectivism, in contrast, is a primacy-of-existence philosophy. For an overview of the "Primacy of Existence vs. the Primacy of Consciousness," refer to that topic in The Ayn Rand Lexicon.
From a comment by the questioner:
My point is that there is no "phenomenon of individual ego", the ego is not a phenomenological existent. What are the genetic roots that validate the concept "ego"? They lie in grammar, not in existents.
The genetic root of "ego" is "consciousness," specifically human consciousness. Consciousness is not an entity or material thing ("phenomenological existent"), but a readily observable faculty or attribute of certain living entities. The questioner, however, apparently denies that consciousness is an attribute or faculty of individual entities. This raises two possibilities: (a) denying that consciousness of any kind exists at all; or (b) an implicit premise that consciousness is a metaphysical absolute, existing separately from any concrete living entity. To discern which possibility the question probably assumes, consider the opening sentence in the original question:
The practice of meditation breaks the cognitive illusion that most of us live with, i.e,. that we are egos, separate selves, something to which the pronoun "I" can refer to [sic], riding around somewhere inside our bodies.
This reference to meditation sounds like classical mysticism of the supernatural kind. The purpose of "meditation" is to dull the mind's connection to secular reality so that it can "contemplate" an allegedly higher "stream" of spiritual consciousness and psychic "energy" allegedly flowing everywhere independently of any concrete existents. Ayn Rand identifies the psychological perspective of such mysticism as follows (from the Lexicon topic mentioned above):
The source of this reversal [of existence and consciousness] is the inability or unwillingness fully to grasp the difference between one’s inner state and the outer world, i.e., between the perceiver and the perceived (thus blending consciousness and existence into one indeterminate package-deal). This crucial distinction is not given to man automatically; it has to be learned. It is implicit in any awareness, but it has to be grasped conceptually and held as an absolute.
The process of breaking one's cognitive tie to secular reality in order to "commune" with a "higher spiritual dimension" also illustrates the ease with which the "D2" mode of cognition (in the DIM Hypothesis) can transition into "M2." D2 refers to cognitive disintegration in its purest form, undiluted by M or I; M2 refers to cognitive misintegration in its purest form, undiluted by D or I. ('I' refers to integration of mind and body, thought and reality, looking at secular reality and striving conscientiously to conceptualize what one observes). D2 originated most fully with Kant, as a systematic means of breaking man's conceptual connections to anything concrete. (Refer to "Kant, Immanuel" in the Lexicon as well as in the index to The DIM Hypothesis.) D2 clears the way for M2, and M2 promises to fill the fundamental human need for integration of some type, to make sense out of chaos, even if the "sense" has no basis in secular reality. The end result of an M2 takeover throughout a culture is massive destruction and death, with chronic pain and suffering for any who manage to hold on under the stifling weight of a miserable secular existence.
Update: Necessary vs. Contingent
An additional comment raises the issue of the necessary-contingent dichotomy. For a very direct rebuttal, refer to "Necessity" in the Lexicon. Additional discussion can also be found in ITOE 2nd Ed. (look under "necessary-contingent dichotomy" in the index). For the Objectivist view of "Absolutes" and "Context," refer to those topics in the Lexicon. In Objectivism, all absolutes and all knowledge are contextual. Understanding what that means is a challenge in the context of traditional philosophical doctrines such as the necessary-contingent dichotomy.
Update: Meditation vs.Introspection
Comment from the questioner:
I see no contradiction between Objectivism and the practice of meditation. Meditation -- at least in the Buddhist tradition -- is mental training of the faculty of consciousness. It has nothing to do with the supernatural.
Well, Buddhism, qua religion, certainly qualifies as mysticism, which Objectivism opposes. But if all that the questioner actually has in mind is the value of mental relaxation and introspection as a means of becoming more "in touch" with one's own subconscious, then Objectivism strongly supports it (i.e., encourages introspection), and it is not an instance of "individual ego as an illusion," which was the original theme of the question (unless the questioner intends "ego" to refer to a false sense of self-esteem, which honest introspection can help one to rectify).
Incidentally, my own primary exposure to "meditation" came in the 1960s, with popular fascination with "transendental meditation." Refer to "Transcendental meditation" in Wikipedia for further description. Consider why its advocates referred to it as "transcendental." Transcending what? The questioner's own description in the wording of the original question certainly seems to confirm the "transcendental" interpretation of "meditation" and the denial of ego as real.