Of course they can. Not only can you just fire your CEO if he says something racist, sexist, or generally embarrassing, but there are non-disclosure agreements

Of course they can. Not only can you just fire your CEO if he says something racist, sexist, or generally embarrassing, but there are non-disclosure agreements (you promise not to tell anyone what happens in the last episode of Game of Thrones), non-disparagement agreements, codes of conduct that entail things you can say. Conan O'Brien had to agree not to appear on television, and not to say anything negative about NBC as part of his leaving the Tonight Show. Nike can tell its employees they're not allowed to wear Puma clothing.