I've been thinking about this a lot lately in calling and sending emails to my government representatives, attending marches and protests. By definition, these are done in public; there's no point otherwise! But there's nothing preventing my employer from deciding they don't like what I'm putting out there and doing .. something about it.
I don't see anything to be done about this, certainly not in the short term, but it's a very different sort of thing than supposing there's something on social media I would be "embarrassed" about my employer seeing.