This first amendment says the government can't abridge your speech. The government is not doing that here, so mentioning the first amendment adds more confusion to the concept of free speech than it adds clarity. It's about as useful as mentioning any of the other amendments.
If customers were actually in legal peril for writing honest but negative reviews of businesses on Yelp, then that would be the government abridging their speech. The court system is part of the government.
This message's purpose is to educate customers that the legal threats businesses are making in those situations are idle ones. They're idle ones because of the first amendment.
No, businesses are saying that it's libel or slander or something else, and if you responded to their suit by saying anything about your first amendment rights, your be laughed out of court.
Yelp's message should be about libel and slander and things like that.
The business sued for libel. The customer's lawyer filed a motion to dismiss, citing first amendment rights. The business dropped its suit. Nobody's laughing.
The business filed a second suit, this time alleging the customer violated a non-disparagement clause in a contract with the business. The customer's lawyer filed another motion to dismiss, citing their first amendment rights. That one's still pending, but nobody's laughing.
Here's a copy of the ANTI-SLAPP motion to dismiss the second suit. SLAPP laws exist to ensure the courts are never used to impede your first amendment rights via the cost of mounting a legal defense. Showing that the defendant was exercising their right of free speech is a required element of making that motion, so it's certainly not going to provoke any laughter in the courtroom.
