As a former FA turned res agent turned furloughed employee that now works for a very large health insurance company.. I can tell you with certainty that by asking for your doctor to reveal what you were sick with, US is violating in a large way HIPAA (Health Insurance Portability and Accountability Act). If they persist in knowing why and what the sickness was then you should contact a lawyer as well as ask your doctor who it is in your area you should contact in regards to violations of that law as it differs from state to state.
On another small note.. I normally agree with Bob on most issues but did want to comment on this one. If people had not taken advantage of the bereavement and medical emergency fares, airlines would not have to "grill" passengers on the issue. I can remember quite vividly a call I got close to Thanksgiving of 1998 where a gentleman that sounded about college age was trying to book from LGA to MIA on a bereavement fare. When I apologetically told him none were available and quoted the best fare we had, he replied; "Well that sucks, my grandma said if I told you she was dead you''d give me a cheap flight". I realize that this paragraph was a little off topic and I apologize, but did want to voice my feelings on that.
Back to the sick pay issue, fight it long and fight it hard. They will get tired of it eventually and cave in one way or another. It was brought up about FMLA and HIV. Well spoken from experience, a "friend" was in just that situation. The things they put the person through were unbelievable. They even wanted notes and test results from 4 different doctors in addition to the company appointed doctor, but good luck not missing work to get that done. At one point the HR rep (which was really just a res sup because they took turns each month being "HR") advised them they needed to submit to random blood draws?! The list of crazy crap they wanted and tried to do goes on, but in the end it took a lawyer and a lawsuit to be filed before they granted it. Hopefully it doesn''t get to that point for you.