I'd say no, I'm no expert or anything but your body needs energy/nutrients etc from food, right? I don't like the idea of fasting (I know that's not what you're saying but since it's been brought up), even if you can survive for x amount of time without food there's a reason you get hungry, your body isn't designed for fasting. And doesn't fasting have negative effects on your digestive system? I could be wrong about that.
On the sickness thing though I know there's a saying, feed a cold, starve a fever. Not sure if that's really applicable to you though
But yeah basically unless you have something where eating would be detrimental to your health for whatever reason, then I'd say to try and eat whether you feel like it or not. Eating food is like the most basic thing needed for being healthy, right?!