At what point did John Nash's behavior change from being eccentric to showing active symptoms of schizophrenia?

That depends on whom you ask, but in Nash's autobiography for the Nobel he says that he did not become dangerously irrational and delusional until his early thirties, when he was working as a faculty member at MIT. Research on schizophrenia in general suggests that he probably manifested symptoms a good deal earlier than that, because most people with schizophrenia begin suffering delusions or hallucinations some time in their late teens or early twenties. This...

That depends on whom you ask, but in Nash's autobiography for the Nobel he says that he did not become dangerously irrational and delusional until his early thirties, when he was working as a faculty member at MIT.

Research on schizophrenia in general suggests that he probably manifested symptoms a good deal earlier than that, because most people with schizophrenia begin suffering delusions or hallucinations some time in their late teens or early twenties. This would also fit fairly well with the timeline of the film A Beautiful Mind, though I should say there were some notable inaccuracies in the film, most glaringly the depiction of his hallucinations as visual instead of auditory (though for that I can forgive them, given the limitations of cinema) and the total exclusion of any reference to his very probable bisexuality, depicting him as exclusively heterosexual (which is not nearly so forgivable, and constitutes a classic example of bisexual erasure---though to be fair, Nash himself has always been cagey about his relationships with men during his youth, which if they were indeed sexual would have been illegal at the time).

In general, it can actually be quite hard to tell the difference between ordinary eccentricity and the onset of mental illness; two different psychiatrists might well disagree on the diagnosis of any given individual's behavior. For some, melancholy is just youthful angst; for others it is the onset of depression. For some, talking to himself was just Nash being Nash; but for others, he may already have seemed to exhibit schizophrenia.

In any case, he was not formally diagnosed until his thirties, and it didn't seem to cause him much impairment before then, so I'm inclined to believe his own report that his symptoms didn't get very bad until then.