Everyone is allowed to believe anything they damn well please. They are even allowed to state it. There should not be a viewpoint holding which makes one unemployable. because as soon as that exists, there is no freedom of speech, only a caricature of it.
freedom of speech is not about saying things that everyone agrees with. It is about saying things everyone disagrees with. and yes, speech comes with consequences. But those should be doled out logically, not by an angry mob forcing an institution's hand.
The common argument that I hear is that free speech does not come free of consequences. Fine. But those should be clear and well defined. Not decided by a mob at any given moment. If you want to clearly state that you will not employ anyone who holds the following views, that is okay with me. But firing somebody because the mob demanded it, is against the very idea of freedom of speech