You have to see this with two lenses. Free speech is always about freedom from government control. Your employer can put whatever restriction on your speech that they so desire even outside of your workplace. In that sense famous people do have less freedom. For example basketball players promoting Nike can't wear any other shoes publicly.
Second case is of government putting those restrictions on you. Can public school put such restrictions on students ? I do not know and I am not familiar with this case.
Second case is of government putting those restrictions on you. Can public school put such restrictions on students ? I do not know and I am not familiar with this case.