And they have the right to fire him. Freedom of speech, doesn't mean freedom from the consequences of that speech.
If it were the private sector, I'd agree with you. This is a public sector job. There's a HUGE difference.
Except when the employer is the government. This is why private schools can limit, restrict or entirely prohibit what students may say, wear or write in class, and public schools can't - at least not to the same degree.
Students enjoy pretty robust freedoms of speech in public school (no, not as robust as an adult per the Court), but private school has NO OBLIGATION to extend those same 1A rights to their own students. The case law is ample in this regard. Government, city/state/federal MAY not infringe on the civil rights, to include 1A rights of their employees.
Pickering v. Board of Education, 391 U.S. 563 (1968) is the case that set the precedent for which all subsequent cases are heard today. Pickering is clear, unless statement made by the employee is "recklessly false", then the school may not "retaliate" against the employee for speaking out on important public issues.
After thinking about this more last night, I think the guy clearly has a claim, and it will be interesting how the court rules. It's actually a perfect case to test the comment that Breyer made just yesterday.