People have the right to express any opinion they choose, but as an employee they also agree to obeying certain policies, such as not speaking to the press, and not posting negative content about ones employer. For example, even when I was part owner of a fairly large clinic, I was contractually restricted from saying certain things about my organization online, and if I did, I would have certainly expected repercussions.
I think there are some things which companies, including healthcare organizations, may do which are so egregious that going straight to the public is both ethically warranted and necessary, However, in this particular case the issue seemed to be over a lack of adequate testing-which isn't just an institutional problem but a statewide problem-and a policy decision on the best place to test.
Regard the former, it's unlikely the institution has any control over the number of tests, and regarding the latter, the location of CoVid testing might need to be based on other issues, for example, would screening outside delay the care for other, more urgent patients, such as MIs and strokes, and trauma?
Now, for me personally, if I disagreed with an administrative policy-, likely one which the org medical director had likely already approved -I probably would first try to change it from within the org, and if that was ineffective, and I felt strongly enough about it, I could in theory, bring it up directly to the State Department of Health or OSHA, where I likely would have significant whistleblower protection, and likely would be in compliance with the contract I signed. Or, alternatively, and more likely-if I felt that my employer and I had significantly different approaches to care, I would simply move on.
But I just think for the most part, negative social media commentary is not the appropriate way to effect change against ones own employer, and likely will backfire, as it looked like it did here. Anyway, I am speculating here, It in this physician's case, it's possible he was putting his contracting employer into a breach of contract with the hospital, and after a warning, and a request to remove his posts was declined, his employer, the contracting org to the hospital, likely had no choice but to let him go.
So, the key point I am making is that although we have rights as citizens, we also have legal responsibilities when we choose to sign an employment contract, and we should only choose to break our contractual obligations in the rarest of circumstances, and when we do, we should expect repercussions. Often, I think we would find there are a lot of other options besides posting to Facebook.