I would wager that almost every educated person in China knows exactly what happened at Tiananmen Square. Certainly every chinese person I've ever really gotten to know knew all about it. The politics around it is complicated, but there is an acceptance of sorts that there's some items the government does not want dwelled upon, and that's one of them.

Don't underestimate Chinese political sophistication, especially amongst what we might call the middle class (a <10% minority in China). There is a common feeling, if not outright belief, that a strong government is necessary to hold the country together, especially during its current transition period with its massive inequalities. I am no expert but my impression is that the people who do know - the middle class educated, with internet access (firewalls are trivial to get around) understand or at least play along with the idea that from a stability point of view, some information is best not fully shared.

I see some interesting parallels between Chinese political censorship and the debate about the NSA revelations, by the way. Both are about concealing information of great public interest in the name of some alleged greater good. The only real difference is that the events in Tianenmen Square happened outdoors.