Not generally, no, but if there was something specific concerning me, maybe? I'm also an emetephobe, so get a bit panicky if I hear of a sick bug doing the rounds.
And the secret? I don't know, it's just who I am, but my parents are also, and always have been, low key about health issues, either just living with it, getting over it, or getting it dealt with, and I'm the same. I also caught every cold and flu going when a child (moved continents, so needed to rebuild local immunity), so getting a fever or whatever, doesn't worry me - it'll pass. My parents and I (and DH and now DC) just don't particularly like fuss.
More practical steps - I eat healthily much of the time (but don't deny myself anything), I maintain a healthy weight, I exercise regularly. Thankfully! I'm generally well, and my parents come from good gene stock - lots of long-lived lives.
I have a few friends with health anxiety, and I really feel for them, it sounds so mentally exhausting, must be so difficult.