The trouble has to do with public funding.
Can a Republican in good conscience accept tax dollars as an employee of a state university?
Republicans have no problem driving on highways funded by the public or for that matter a multitude of public projects.
Of course.
--Can a Republican in good conscience accept tax dollars as an employee of a state university?
Sure. Just make a deal with yourself before you start to tell only the truth. If you get fired or denied tenure, so be it. The Lord will still provide for you!