Speech does have consequences, but the university also needs to tread lightly and remember that it is a government organization. Will be interesting to see how this plays out here at my school.
Yes, he most certainly has a case, I feel. I seem to recall at least one Supreme Court case from decades ago (Pickering v Board of Education?) in which a teacher was fired for unpopular speech made in their capacity as a private citizen, and that the SC ruled that teachers couldn't be fired for such. Obviously the professor in question came with a pre-existing social media following and a resumé with several previously-published books, so the school can't really claim that they have any possession or right to regulate the social media account nor that it was reflective of anything other than free speech made as a private citizen.