My dad has this disease and so do i and it actually helps the immune system to a degree. Ur blood cells use iron to help fight off disease and other things, since ur body stockpiles it, it should always have a supply. I seriously can't think of anything about haemochromatosis that would affect the immune system, do u play a wind instrument or have a retainer? If so make sure u clean both of these once u get better from the illness, because i kept getting sick for a few months and thought it was the same thing, turns out i had small amounts of mold in the neck of my sax, after a good cleaning i was feeling much better again. I found denture cleaner to work the best.
I knew someone with this condition who occasionally had problems but who regularly needed to give blood. Her main issue was with fatigue but she managed to compete in an Olympic final.
I don't have the answers but I would make sure you don't over do it while you are sick so that you can recover quicker.
J