Can a person sue a site for violating one's right to freedom of speech?
Not sure if you're asking seriously or not, but my unqualified opinion is that, yes, a person can sue for anything. But that they wouldn't have much luck with the 1st Amendment angle. The First Amendment only forbids governments from impeding ("abridging"?) free speech.
A message board site, can, if it wants, decide that everyone who never used the word "purple" at least once on the site can no longer post on the site. And now I'm covered if that happens.