[Timpanagos1]

“Private school...with public money.”

Yes, but they are able to control speech and other displays of a political, social or religious displays on their campus.

[FredZarguna]

The case law, decided by the currently sitting Supreme Court is that public universities may place reasonable restrictions on time, place and manner, but such restrictions are to be examined under the standard of strict scrutiny.

That severely limits the ability of public universities to limit free speech, or any other Constitutional protection incorporated against the states.