It's tiring to see people make the case that because the government was not involved, that it's by definition completely unrelated to free speech. Like, you clearly know that's not the case, from this follow up.
Of course the right to free speech does not extend to compelled speech, as you say.
ACLU defended the Nazis because they support free speech. But "free speech" of course doesn't compel them to. But clearly if they had chosen not to, they'd show they care less about free speech than they say.
Like the ACLU of today, who have said that they'll only defend speech they agree with, or something to than effect. Well, morally, do you actually have that value anymore?