I'm not an expert but it doesn't really seem like a first amendment issue although I think most would agree that the government shouldn't be punishing its own employees for conduct like this.
https://en.wikipedia.org/wiki/Pickering_v._Board_of_Educatio... establishes a "right to speak on issues of public importance" for public employees.
https://en.wikipedia.org/wiki/Garcetti_v._Ceballos limits that right when statements are "made pursuant to his position as a public employee".
https://en.wikipedia.org/wiki/Demers_v._Austin extends "First Amendment protection to professors at public universities for on-the-job speech that deals with public issues related to teaching or scholarship, whether inside or outside of the classroom", but doesn't apply everywhere as it's a Ninth Circuit decision.
I think https://en.wikipedia.org/wiki/Demers_v._Austin has it right in this regard - that public educators have a specific First Amendment need/exception to be able to teach facts their chain of command doesn't appreciate - but it doesn't yet apply in Texas.
It will be interesting to see what effect these kind of antics have on quality of higher ed in these states. I must think that there would be some effect on the quality of professors that get hired.
https://www.thefire.org/news/judge-university-florida-cant-e...
Professors at state universities are state employees and you can't fire a state employee for criticizing an elected official.
Besides, IMO tenured professors are the very front line of free speech, perhaps right behind journalists - for the system to function at all (/continue limping along), tenured professors should NEVER be afraid of criticizing the government. At all. In any way.
I don't think she's tenured — her TAMU Web page says she's a clinical assistant professor. [0]