If you want to exercise your rights, you are free to do so on your own time. In this case, what's being said here is that this officer is never allowed to exercise his rights on the job or off. So there is a big difference there.
I have the right as an American citizen to vote. I cannot stop at the polling place on company time and vote. I can vote before I go to work or after, therefore the company is not interfering in my right to vote, just not on their time.
what you do not have the right to is a job, so if the people paying you have the right to fire you if they do not like the way you exercised your rights.
This has nothing to do with rights to have a job. It has to do with rights to free speech.
But his free speech was not violated. The first amendment guarantee for free speech is that no LAW shall infringe it. And even that’s not without limits (threats, incitement to riot, perjury, plagiarism slander, fraud).
The officer was not imprisoned or arrested. No violation of his free speech rights. He was fired, which is the right his employer, even if the employer is the government.
Again: the right to free speech is the right not to be fined, arrested, imprisoned, or otherwise found in violaion of the law. It is not a guarantee that you can’t be fired, or denied a platform for your speech