What happened to this woman's first amendment rights?
Well, the First Amendment protects the right of free expression of an individual acting as a private individual. It doesn't protect people who make comments on the authority of their position -- an employer or organization may place limits on an employee or officer acting in her official capacity (and therefore representing the organization).
Also, if the position is basically a political appointment, then the appointers can basically terminate her position for any reason that they want.
If it's truly good and powerful, it deserves to engender a thousand misunderstandings. -- Ben Ratcliffe