Quoted from a lawyer who specialities in law as it relates to labor and employment:
A common misconception is that employees have a constitutionally protected right of free speech in the workplace. The truth is, public sector employees enjoy some limited protection. But if you're a private employee, the First Amendment will not protect you from being fired for something you say at work.
Likewise, it won't prevent your employer from firing, disciplining, or demoting you for things you post on social media in your free time. This is true even if your statement is not criminal, vulgar, or offensive.