Well, "olmak" is the culprit here. If you are talking about being sick at a specific instance in the past, you would have to say "Hiç hasta değildim." With "olmadım" though you are making a general statement about your past.
"Olmak" is literally "to become" so here you are saying something like "I have never gotten/become sick." :)
It seems to me that hiç is some kind of connection word, as you see it in hiç bir (no one), hiç kimse (nobody), and now we see it as hiç hasta (never sick). The negative verb is what is probably confusing everbody because it seems to enforce what is already stated. It could be that hiç and asla have distinctions for instance hiç son = never ending, and asla son = never last. We have to bear all these in mind as we go through Duo. After all, Duo is a good stepping stone to aspire from.