You can say what ever you want in a contract. The problem comes when you try to enforce them. Here what can the doctor do? The are not allowed to disclose ANY patient information, including the fact that a certain person is a patient. In order for them to send a take down order they would have to acknowledge that the person posting the comment was a patient.

Sign the agreement, post something nasty and if they issue a take down or suit - sue them back and report them to the medical board. I don't think this will last too long.