The perplexity occurred as it has two meanings but if I read between the lines, they are contradictory to each other! (Let's not touch adjectives of this, they are even more confusing!)
And the next meaning is...
valetudinarian (n; #2): a person excessively worried (anxious?) about the state of his health; [
hypochondriac-omitting this as this is the medical term and being a healthcare provider, and though it's close, I feel it does not exactly represents our word in concern.].
Now, if Jack is too conscious, worried or anxious about his health, he's less likely to contract diseases. We often say, he's, though over, health conscious. On the other hand, the first meaning says that if you are a valetudinarian, you are chronically sick (one of the meanings).
If you worry too much for your health, it is actually to prevent valetudinarianism, isn't it?
How come a valetudinarian (#2) person can be a valetudinarian (#1).