I have a NON VEG EATING KID. I love 'er, but this part of her behavior drives me to distraction. Kid #2 is a veggie hoover, however.
I would say focus on getting healthy non-veg foods in your kid's lunch. We go for
1) savory: sandwich w cream cheese and turkey; PB & J; or hummus and crackers (which she dips herself)
AND
2) sweet: banana, apples (more likely to be eaten if sliced), applesauce, berries, low-sugar fruit salads; oranges (most likely to be eaten if sliced in the round or peeled segments), or fruit yogurt
AND
3) "snack": popcorn (unsalted, home-popped); cheese slices/sticks
At least that way I know she's eating stuff that is more or less healthy...
At dinner time we offer dessert only if a certain amount of healthy veg has been consumed. Dessert is usually something like a marshmallow, gummy bunnies, one scoop ice cream, chocolate banana pop. Enticing, right? Well, I guess she super duper hates veg, because she often goes without.

DD#2, however, shovels everything in.
