Exercise and Diet when ill

I am always torn when I take ill in any fashion, uncertain of how to handle my dietary and exercise regimens during this time period. Part of me wants to just veg' out, rest up, and eat up to help give my body all the fuel it needs to get healthy again, while another part of me always wants to power through it. Does anyone have any facts about whether continuing to operate in a caloric deficit could impact my body's ability to recover from illness in a speedy fashion? How about exercise, which is obviously going to put a moderate strain on my body and my metabolism (that's the idea, after all). I don't have any sort of flu-like symptoms, so my digestive processes don't seem to really be affected, but I am still fatigued and achey and my sinuses are a disaster. How do you recommend I handle things?

I should probably also mention that I've been STARVING lately. Like, for over a week. No matter whether I've been sticking to my diet properly, or eating up my calories by having a few drinks, or going over my budget and having to exercise things off. I just can't seem to get satisfied nutritionally.