Not the op, but no - I don't see anything wrong with the scenario: the employer is making the call, and if they find the speech of the employee doesn't fit with their worldview they have all the rights to fire them.

Practical example: the employer is an LGBTQ+ friendly establishment, the employee is on social media saying that LGBTQ+ people are all deviants and will all burn in hell for their sins. I think the employer should have the freedom to fire the person, right?

Forcing the employer to keep the employee is the equivalent of compelled speech.

Edit: fixed - no joke - pronouns