A problem I frequently encounter in my practice is children turning their noses up at all vegetables, perhaps with the exception of raw (not cooked!) carrots. Parents feel this is really not healthy for their child, and to some extent, I would agree. However, raw fruit can cover a lot of these bases, so feel good about that if your child will eat at least 3-5 different fruits in the amount of 2 or more per day.
I am not going into all the nutrients that vegetables provide, but I will address a few. They are a source of the vital mineral potassium, one that is in short supply in most of our diets. Potassium is especially important to balance our sodium (salt) intakes, as they are complementary nutrients, working to balance each other's effects. Some experts feel higher sodium intakes are not a problem as long as they are accompanied by good intakes of potassium, and I would concur. Fruit is an excellent source of potassium, and an excellent source of vitamin C when eaten raw or very lightly cooked. You can also get a lot of "phytochemicals" from deeply colored fruits like berries; these play antioxidant roles in the body.
But should you resort to preparing vegetables in any way that will get your child to eat them?
That was the suggestion of Rachel Ray today on her television cooking show. She was making tempura with her guest Carla from "The Chew." Rachel said "If your kids hate vegetables, put some batter on them and fry them." This poor nutritional advice was made even worse by Carla's recommendation to use vegetable oils, canola, sunflower, rice bran oil, to fry them in as they have a higher "flash point" - I think she meant "smoke point." This is actually not completely true: https://whatscookingamerica.net/Information/CookingOilTypes.htm (To her credit she did recommend refined coconut oil - much better choice, but a pretty expensive way to fry.)
What is wrong with this advice? Foods fried in industrial seed oils are some of the worst foods for health, foods I recommend my clients limit because they contain numerous dangerous by-products of highly-heated polyunsaturated fats. A great article to read on this: http://www.westonaprice.org/know-your-fats/the-big-fat-surprise-toxic-heated-oils/
Some researchers feel that these industrial seed oils, a relatively new introduction into the human diet, are at least in part responsible for the obesity epidemic, the increase in cancer, even the rise in blood sugar control issues. Fats become part of our cell membranes, which are the "gatekeepers" of our body's metabolic machinery. Hormones dock on the cell membranes, sending messages that regulate all aspects of our metabolism. If the docking sites are not operating at full capacity, we can may become slightly resistant to the effects of the hormones. The presence of "damaged" fats can also initiate oxidation chain reactions - leading to damage of cellular organelles, like the super-important mitochondria, which generates the majority of our body's energy.
Vitamin E in adequate amounts can help limit the damage, and vegetable oils are considered the best dietary source. However, the oxidation of fried oils will exceed the protective effect of any vitamin E that is present in the oil, so a net deficit of vitamin E actually can occur.
So do your family a health favor, limit fried foods, and if you choose to fry, do so occasionally
Try to use a more saturated fat such as beef tallow. I get my own tallow from the surface of beef bone broth I make. I lift it off when the broth is chilled, rinse with cold water, pat dry, and put in your freezer. Just be careful and heat slowly as the traces of liquid can lead to sputtering when you place it in your frying pan. Beef tallow has a high smoke point of 420 degrees, http://cooklikeyourgrandmother.com/how-to-cook-with-beef-tallow/, which you should not approach if you keep the temperature around 350 degrees. If you don't use a frying thermometer, just make sure the oil does not smoke, which means you let it get too hot.
If your children like chicken nuggets and French fries when you go out to eat, try to limit this meal choice to 1-2 times per month. You can try making baked chicken fingers, a good recipe is here: http://www.recipetineats.com/truly-golden-crunchy-baked-chicken-tenders-less-mess/. I often made something similar to these when my kids were young, using corn-flake or Panko crumbs and a mixture of olive oil and ghee, tallow, or palm oil to coat the pan (the recipe here says brown the panko crumbs in oil, I never tried that but may be a good idea.) Finally, good sources of vitamin E are fresh nuts and seeds, avocados, and cold pressed olive, sesame, coconut, and almond oils used in salads or in a lower temperature saute.