People have different reasons for being shy, and react differently to different forms of speech. The best phrase to make someone comfortable would probably vary with the person, setting, relationship, etc.
That said, what usually works well for me is not to ask them whether they want to eat, but to give them a choice of food right away - "Hey, let's have some chips and salsa. You like mild or hot?" Whichever they name, they'll now feel like they were part of the decision, and will be less shy about eating.
If this doesn't work and they instead say something like "oh, I'm really not hungry," then don't get too much on their case. Maybe they really aren't hungry. Or maybe they have some reason for not eating that they'd rather not talk about. Let em be.