Ask the person why they don't eat. Maybe they don't like the place... Let them pick the spot next time maybe. Tell them that it makes you feel bad and awkward when you guys go out to eat, but they never eat.
You could always do something else together that doesn't involve food.
You could instead ask them why exactly they never seem to eat. I don't think you should feel rude doing this, either; I mean, it's a simple observation. Just be genuine and honest with your concerns. Tell them it makes you feel uncomfortable.
Next time, say you feel awkward eating by yourself, so you'll only go if they eat.
My friend let them not going with you again to eat.
'If you don't eat, I'm not going'