What first amendment right exists on Facebook? Other than Mark Zuckerberg’s? 

There are no first amendment rights on this forum. Nor in the New York Times. Nor in the Bayside Gazette. 

From Chandra Law:

May I sue Facebook, Twitter, or other social-media companies for violating my First Amendment or free-speech rights?

No. The First Amendment restricts governmentalaction only. 

We are contacted frequently by people complaining that Facebook, Twitter, Intagram, or other private corporate social-media companies supposedly violated their "free-speech" or "First Amendment" rights. These potential clients want us to "sue" these social-media companies for them.

But Facebook, Twitter, and Instagram are privatecompanies. They are not the government. They can set their own rules on speech and there's nothing you can do about it. A court that tried to do something about it would be violating those companies' right to free speech. And conservative judges have ruled—and conservative politicians have insisted—that private corporations have a right to free speech.