I wonder what people are thinking when they try to get fired a TA who might (or might not) have posted offensive crap online. The moment you make this acceptable prepare for “You said something objectionable online so here’s the door” to be the answer whenever you ask for a raise, try to unionize, criticize the administration, etc. And even if you’ve never been online in your life, how easy is it to make it look like you are the most obnoxious person on social media?
Also, if employers have the right to evaluate the morality of workers and fire them if they disapprove, how can anybody make an argument against employers who fire people for using contraception? According to this logic, an employer should be able to fire not even just for actual use but also for writing a pro-contraception tweet.
That’s why I always say: figure out what your principles are and proceed from there. Are you in favor of expanding employers’ rights to having control over your social media activities outside of work hours? That’s what’s at stake here.