We all need to develop an understanding of what the First Amendment does and does not protect.
“Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.”
All this does is prohibit Congress from criminalizing speech. We are free to speak our minds without fear of criminal prosocustion. That is all.
The First Amendment DOES NOT prohibit employers, private or public, from disciplining, even terminating, employees for speech the employer finds offensive. There are some additional federal and state laws which do provide some protection in this area but they are not uniform and almost all allow the employer to terminate an employee for speech which, in the opinion of the the employer, brings disrepute upon the employer or misrepresents organizational policy.
The First Amendment DOES NOT prohibit other individuals from expressing their distain, shunning the speaker or applying other non-criminal sanctions. Any individual can speak their mind; but, any other individual can decide the speaker is an idiot and respond accordingly.
The First Amendment DOES NOT prohibit employers from considering an individuals public statements when making hiring or promotion decisions.
So yes, government employees have freedom of speech as defined by the First Amendment. We can say what we want, when we want, how we want without fear of criminal prosocution for what we say.
But there is no guarantee we will have a job or meaningful promotion potential after we have said it.