[_delirium]

I'm not sure it's that they're looking to change it, vs. take advantage of something for their own political ends that already exists (and has for a long time). The traditional "rule" in the US is that you can do whatever you want on your own time, unless it's something that your employer finds out about, and finds sufficiently offensive, embarrassing, or inconvenient. Then, since employment is at-will, they can fire you for it, as long as it isn't from a specific list of things that would constitute unlawful termination. The main change, as I see it, is that using it as an organized tactic (compiling lists, contacting employers, etc.) has mainly been done by conservatives in the past, directed at communists, the LGBT movement, flag-burners, etc. Those more on the left using it (or at least, using it with any success) is a somewhat newer development in the US.

To me the fundamental place to tackle this would be at the mechanism: the problem is that employers can and do fire people who have opinions they either dislike or find embarrassing. That will continue to be true, unless labor laws are changed to make it more difficult.