This is absolutely incorrect. To Americans, Freedom of Speech is, as the person you're replying to said, an inalienable right of all human beings. It's a fundamental principal of our society. The first amendment is a legal constraint on the US government that keeps them from violating that right. It does not grant the right, the right belongs to all humans simply from the nature of being.
Think of it like this. People don't think murder is wrong because it's illegal. We have laws in place making it illegal to setup how government handles murder, and we want those laws because murdering a person is wrong at a deep and fundamental level. The laws follow from the underlying ethical beliefs. Freedom of Speech works exactly the same way.
So a company like Blizzard censoring an activist is not a violation of the First Amendment, but it is arguable a violation of Freedom of Speech.
But in reality you can't hide the fact, that limiting freedom of speech is very much in the interest of those who want full control on what we can say.
A good example is the Muhammad cartoons, which reminds us that there is a very long way to go before religions can exist peacefully.
https://en.m.wikipedia.org/wiki/Jyllands-Posten_Muhammad_car...
So when we see it being limited it freaks us out.
Going with the same kind of example: society agrees that murder, thievery and lying are bad things, yet find it justifiable to kill in self-defense, to steal in order to eat, and to lie in certain situations.
I'll also say that it's very different to limit a certain type of speech than to limit speech you disagree with.