As an employee of a hospital can they post who has and hasn't gotten flu shots. Or can the staff have a list of people that have not to enforce them to wear masks?

Is there a requirement that you have flu shots? I don't know the purpose of publishing a list.

If hospital policy says "no flu shot, you wear a mask" and you don't have a flu shot they can post a list so the charge nurse (or someone else) is aware that you need to wear a mask.

My husband was a patient in ICU for a lengthy period and this argument raged at that time. I did not want a nurse who did not have the flu shot in his room unless she wore a mask.

That, of course, was hospital policy.