MLB facial hair rules are going to be specific to each Club. Whether a player has a beard, mustache or a different hair style, organizations and Clubs are in control of their own facial hair policy.
Yes, facial hair in the workplace (even on the baseball field) is an employment law issue. In 1976, our Supreme Court affirmed beards and goatees could be controlled if associated with a government position or job.
In other words, most baseball players are free to make a choice specific to their hair style. Of course, the exception to this are players associated with the Yankees and Marlins.
Like in most cases, if you are playing well, your hair will likely never come up.