“Question: Why is this a Constitutional issue?
Is the university a government organization, under the jurisdiction of the federal constitution?
Or perhaps it is a State institution?
(And, if it is part of the government [federal or state], which branch does University fall under?)”
IT’S BECAUSE THEY TAKE FEDERAL MONEY!
For example, local Courts cannot compel a defendant to testify against himself. The rule holds in all Courts, not just Federal Courts.
Likewise, I don't think that the State government can abridge the freedom of lawful speech. I believe that expulsion from the University of Oklahoma based purely on socially objectionable speech is unsustainable.
So then, are recipients of, say, SNAP constrained by the federal Constitution?
What about civilian contractors? Are they bound by the constitution because they receive monies from the federal government?
If not, why not?