You can't fire them for the speech - which is of course protected - but you can fire them because they can't possibly be qualified to hold their position while holding/espousing those views. Unless there's a specific state law protecting employees from off-duty conduct (Pennsylvania doesn't have one), employees can certainly be fired.
I would think that she would want to prove that statement falsely attributed first in a defamation suit before going after the university.
The test above is correctly quoted, the only dispute is factual which none of us here has the evidence to resolve. She may be lying or telling the truth, but we can't make any grand pronouncements about whether the university was right or wrong on the basis of disputed factual claims that have yet to be resolved in court.
And it's the one comment she says she didn't make.
I wonder if her lawyer has seen this story? Many of the comments here could be helpful in showing damages.