The First Amendment doesn't guarantee you the rights you think it does

If you work for a private company, it’s probably not a First Amendment issue.

“It’s the company’s right to discipline their employees’ speech,” Nott says.
But just because a private employer has the right to fire someone for something they say doesn’t give them legal carte blanche. Depending on what the fired employee said, the employer could be in violation of the Civil Rights Act, or possibly in violation of contract law

If you’re a government employee, it’s complicated.

Institutions like police departments, public schools and local government branches can’t restrict employee’s free speech rights, but they do need to assure that such speech doesn’t keep the employee from doing their job. It’s definitely a balancing act, and the rise of social media has made it harder for such institutions to regulate their employee’s speech in a constitutional manner.