No, there is no legal requirement for anyone in public office or running for public office to share his/her health history. But revealing the basic details of that history (e. g., Rudy Giuliani and Fred Thompson both being forthright about prostate cancer) helps a candidate create a trusting relationship with prospective voters. This is especially important for presidents and presidential candidates since the impact of their health problems on their job performance is of most import to the public.
Slick who didn't want the doctor to know what the clear fluid was ... said it was for allergy.
IIRC, that's because the doctor at the WH at the time was a Bush administration holdover, and "Slick" didn't trust him with the info. In fact, I believe that WH doctor was forced to resign shortly thereafter. Doctors should not, as a matter of ethics, discolose their patient's medical info unless the patient specifically permits them to do so. So Clinton does have a legitimate privilege that he exercised to not let us know. Other presidents, BTW, have disclosed even serious medical info (e. g., Eisenhower, who suffered a heart attack and a stroke) while others tended to keep things under wraps.