The US first amendment generally protects people making statements of opinion, and statements of fact, even when these are clearly or allegedly false, unless they constitute defamation or fraud. Here the statements mentioned in the question and reported in the linked article do no seem to be about any particular person or group, so they are not defamatory. There is lots of precedent for this.
Federal and state anti-discrimination statutes only protect specific classes of people. "vaccinated" is not such a class, surely not on the Federal level, and I do not believe it is on the state level in Florida (or any other state) either. Thus no anti-discrimination law could prevent a private school from taking the action reported, restricting or firing teachers who hae been vaccinated.
Being vaccinated is not a "disability" under the ADA, and so the ADA protections against discrimination on the basis of an actual or perceived disability would not apply.
An employer may not, under HIPPA, demand medical information, such as vaccination status. But if a teacher reveals it, or the school learns of it in some other way, they can choose to act on it. Florida is, like most US states, an "employment at will" state, so unless an individual has a binding employment contract (which most do not) an employer may fire an employee at any time for any reason or no reason, except only those few reasons prohibited by Anti-discrimination laws