That is not what free speech means.
Free speech means the government cannot prohibit free speech. A private institution can take any lawful action they want in retaliation/reaction.
I agree that it really sucks that saying something true can get you fired, but this isn't an infringement of the first amendment.
You're right. I was being very Ameri-centric. I subconsciously interchange free speech and the first amendment even though they are not equal.
I do believe that individuals and private institutions should have this right to react though. I don't agree with how it was used in this situation, but I absolutely believe the hospital should have the right to terminate someone based on the opinions they openly share.
If this same employee was sharing an anti-vax opinion I would want the hospital to be able to remove them from the role.