Indeed, allowing employers to coerce the speech of employees by punishing them for actions outside of the reasonable scope of employment is very worrying. Unfortunately, a good deal of the modern "left" supports it when they dislike the person. Much of the right does too, which I have equal disdain for, but I will at least acknowledge it can be logically consistent with some right-wing philosophies (on the more ancap end). Whereas it's a bit strange to see "socialists" saying "but they're a private company!"

However speech in the classroom is within the scope of your job duties. So my employer should not be able to fire me for wearing a Trump or Biden sticker off the clock, but it is fair to prohibit me from wearing it whilst on the job, and to sanction me if I'm proselytizing to customers during my duties