I heard you were sick [the person may or may not be still sick] OR I heard you got sick [if the person is still sick]

Yes they are correct,but it depends on what you are trying to ask about as far as other expressions are concerned. If you want to just ask the person up front if they are sick you would simply ask" Are you sick?" If you are just curious about something you may have heard from someone else you would use the two phrases you have there

通常可以说： I heard you were sick. 或者 I heard you were not feeling well. 我还没听见过 ： I heard you were ill.

Either is correct. Different places, and different people in the same place have different preferences on how to say it. It can refer to someone who was sick, but is not now, or it can refer to someone who is still sick, but who you are now talking to. I heard you were sick. and I heard you were ill. The meaning is exactly the same. No difference.