Lemme ask this.
When Neal bans someone from his website, is that an infringement on that person's First Amendment rights? Is Neal breaking constitutional law?
Of course not.
This is a privately owned website and Neal can ban anyone he chooses for no reason than he chooses to do so. I have no First Amendment rights on a privately owned website.
Why would this be any different than being fired from a private company?
Now, depending on the law, there may be isolated circumstances where people are protected from being fired for saying stuff. But that protection comes from another law/regulation and not from the First Amendment.