It would pass a 1st amendment challenge because it has to do with regulations in a place of employment, not in public. Rights can be restricted by employers to a small degree to promote a neutral workplace, such as dress code, profanity, hygiene, etc.

As an employee, you are an extension of the company brand/reputation, thus, must conform to certain rules that the company sets to continue its operation at a predictable and efficient pace. Politics and self-expression destroy that predictability and promote drama and chaos in the workplace.