This is a private company that has decided that the actions of one of its employees reflected poorly on the company, and decided to act. That is certainly well within their rights, and does not infringe on the rights of the employee - the employee is free to express whatever opinions he wishes, but does not have the right to be free from the consequences of those opinions.
Of course, that is much different than the Chick-fil-a matter. In that case, public officials were threatening to use the power of government to punish a company for speech it did not like. That is a clear violation of the First Amendment.
Now, if the university this guy lectures for decided to fire him over it, he might have a case, assuming that it is a publicly-funded university.
He’s an adjunct, not tenure track. They get fired and hired all the time, with the demands of the semester. All they have to do is say his services aren’t needed.
There are adjuncts who’ve been at the same school for 5-10 years and still wake up one day and find they aren’t renewed for the coming term.