I'm sorry but nowhere in the free speech rights does it say that your employee has to abide by it. It's a right that protects speech from the government, not from your employer or from anyone else.
In fact, pretty much EVERY single company's employment contracts will include a vague "you can't bring bad press to the company" clause, which covers social media posts etc.
If the company disagrees with your politics, then yes, they can fire you for it.