I was asked by my HR person what doctor and medicine I am taking for depression.

My sister says that this is none of their business and they really can not ask it.

Just by asking for that information is not a violation. If you don't want to give him that information, tell him it is confidential .

Tick

Why were you asked this question? How did they know you both treat with a Physician and are taking medication?

You certainly can refuse to answer but keep in mind that HR apparnetly has some concerns, has not simply fired you, appears to be attempting to find out if your depression is controlled - and most States are employment at will States which means you can fired for ANY reason at all (including refusing to answer) unless you are protected by a contract or Union.

Is there a problem with your work performance and that has caused HR to ask this question?

But, no, you don't have to answer.