Contract that removes Freedom of Speech?

Yes, they can have you sign the contract. Yes, it is legally binding. Yes, you can refuse, and they will then refuse to serve you.

All sorts of rights there.

"Contract that removes Freedom of Speech?" No. The expression "freedom of speech" expresses the concept of a limit to a government's power over its citizens. Not applicable to these circumstances.

"Can a doctor's office require a patient sign a paper saying they will not make any negative public statements about their office before they are seen by the doctor?" A doctor's office, like any business, has the right to draw its own contracts and attempt to enforce them in a court. A patient, like any consumer, has the right to negotiate the terms of his or her contracts, and the responsibility to enter into only contracts s/he understands and intends to uphold. In a capitalist economy, it is the free market (and not the legislatures, for the most part) which controls the existence (or not) of contracts.

"If the patient signs this paper, is it really legally binding?" Most likely it would be upheld. There is consideration on both sides, and there are no criminal or civil rights violations. There might be administrative law which would apply, and that would vary by jurisdiction.

"Does the doctor's office have the right to take the person to court because they gave any honest review of their experience online?" No. But they would have the right to impose any penalty outlined in the contact (and attempt to enforce that in a court) for the specific act of "negative public statements about their office before they are seen by the doctor."

The same thing that separates Law from Pretense separates Rights from Rites - the 'con-track' you describe is pure Talisman Ritual; If you lied to folks about the doctor raping you during your colonoscopy, you could be sued for giving them a black eye and you would lose withOUT the con' game of that magical spell on paper -AND- WITH that magical spell on paper, if the doctor DID rape you during your colonoscopy, the magical spell on paper would add conspiratorial effort at preparation to prove the doctor's intent to rape preceded the rape so the doctor's lawyers would have all copies of those contracts shredded before you could say "Discovery Motion".

The problem with crap like this is you don't really know why it's being used - You can assume it's because the doctor is incompetent and wants to prevent ramifications - and that might be valid; Equally valid, however, is the simple 'loyalty obligation' of helping support friends and family, where those forms were among the first efforts produced by some fresh-graduate who just got the green-light to "practice law"; so ol' Doc's just "gives the kid a break" and pays third-cousin-twice-removed for the "brilliant legal work" as a 'good kid' 'pat on the head' - The TEST for which case is at work is simply to chuckle to Doc': "You -know- this paper ain't gonna save ya, right?" - MOST Doctors fall into Two categories; "What paper?" because they got no idea what their 'Secretaries' do at all - and my favorite, Laughing WITH YOU and explaining To you just how they happen to be related to the nincompoop 'prodigy' they hadda buy that door-to-door gadget from and they'll crumple-up and dispose of that toilet-paper themselves - If the Doc gets defensive about it, however, you know you're dealing with a cowardly psychotic thief.

Freedom of Speech does not give you the right to slander someone or to say something that will negatively affect someone's business (restraint of trade) so yes. They can.

This paper is worthless.

If something is the truth, then it is not slander or libel.

I would be weary of someone who makes you sign such a paper. Honest people aren't afraid of the truth,