This story once again brings up the question in how much of a reach an employer can legally have on their employees personal lives. The National Labor Relations Board (NLRB) argues that the employee's free speech rights were violated.
People forget that the First Amendment prevents the government from censoring it's citizens, private companies and individuals are not barred from controlling speech. In my research American Medical Response of Connecticut Inc is a private company, not a government agency so the employee has no leg to stand on. My current job has a similar policy regarding all social media that comments regarding my company can get me fired. I personally don't like the policy but it is what it is and I stand by the creed that if you don't like the policy then go and find another job. If you sign the employment contract than your bound by the policies the company puts in place. So before you go off on your boss or a co-worker who pissed you off on your twitter, myspace or facebook, check the policy first.