The thing about free speech is that there is no freedom to say whatever you want without backlash or negative response. If a perponderance of people who have the power to do so decide that they don't like what you have to say, they are under no obligation to keep facilitating your right to say it. At least in the US, even the government still retains a fair number of restrictions on completely free speech, and in the private sector there are even more. That's part of the deal with how free speech works.
In most non-government employment situations, unless state laws provide additional protections, your work has wide latitude to curtail your speech or punish you for it. For example, although a police officer cannot arrest you for wearing a hat with a political statement on it, your boss may well have the right to fire you for it.
It doesn't have to be for a good reason, or a bad reason - just as long as it's not for an illegal reason. Somewhat ironically, this sort of "Right to Work" atmosphere was championed by the Right for many years until it fianlly started to bite them in the ass a bit. Oh well - too bad, so sad....
