It depends on exactly what you mean by Objectivist. If, by that, you mean a person who explicitly understands Objectivism and lives life deliberately by its guidance, then it's unlikely to ever happen that everyone will be an Objectivist. There are many people who are either unable, unwilling, or simply uninterested in the deliberate study of philosophy that would be required. Studying philosophy just isn't for everyone, and that's okay.
However, if by Objectivist, you mean a person who implicitly agrees with and follows the basic tenants of Objectivism without having given it much study, then it is certainly possible that everyone could become an Objectivist*. In such a case, you wouldn't necessarily have everyone quoting out of Atlas Shrugged or making deliberate philosophic speeches, but you would see it be a part of the normal culture that people would earn their own way, act in their best interest, and deal with one another honestly. If you were living in such a society, it would seem abnormal that people ever acted in another way, and tales of how things used to be would seem strange and foreign (e.g., "When I was your age, the government actually made a law saying you have to buy health insurance!" followed by "Wow, that's really weird, Grandpa!")