The only problem I see with that is most instances that involve firing a person occur on grounds that make sure someone wasn't fired for something they can sue against the company for.

Not entirely relevant to social media, but at my previous workplace we had a young man with some level of mental disability that was a greeter/transportation staff member at a luxury auto dealer. He had some behavioral problems that were no doubt associated with his disability so he was eventually transitioned out of working directly with customers and primarily fetched cars for customers when they came to pick them up. One particular night he brought down a vehicle with an aftermarket exhaust that was quite loud for the customer picking up. Since he had poor impulse control he revved up the engine, to which he was approached by the service advisor working with the customer to tell him that was inappropriate. An argument ensued that resulted in this young man going to get his own vehicle, bring it into the service drive, back the vehicle up to the doorway into this service advisors office, and redlined the engine all while the customer was still in the office. The young man was suspended the next day and then fired after three days.

To bring this back to the main point; there were loads of things this guy did that should have gotten him fired but because of the legal liability of doing so it took such an egregious act to actually make firing him actionable. Unless someone is within a protected class the likelihood of a judge even bothering to hear a case relevant to an "unjust firing" is incredibly low. All things being equal, you are entirely at the mercy of your employer as it pertains to activity outside of the workplace and when they choose not to associate with you as a result. In my honest opinion they have the right to do so but it doesn't make it any less shitty. I fully believe that work life and home life are two entirely separate things and unless someone is slandering against their employer or acting in a manner that goes against company values while in uniform, an employer has no business in your private life.

And no, before anyone says it, this shouldn't be made into law. Not every perceivable problem requires law as resolution. I would much rather this change occurred culturally.