L.,
I understand your concern about weight, but here's the thing... not all calories are created equal. If your boys are filling up on refined carbohydrates, they will not be hungry for good food. It is not our responsibility as parents to make kids eat; it is our responsibility to keep good foods available for them when they are hungry. It is not true that it is better for them to eat junk than to eat nothing at all. Rest assured that children will not starve themselves. They probably will go a while without eating, there probably will be tantrums and food thrown while they test their boundaries, but when they figure out that they are not going to get their way and when they get hungry enough, they will eat. Ask any doctor - children will not starve themselves. Catering to their pickiness will only make it worse.
The best thing you can do is figure out what GOOD, nutritious foods your sons like. If they hate green beans, wait a little while before you offer it again. It make take a child up to 17 times being exposed to a food before they are willing to eat it. On the other hand, make sure they have other healthy options if you know they don't like what you are serving. For instance, my 3 year old hates beans. She has since she was a baby. When we eat beans, I still put them on her plate, but I also make sure she has plenty of other food to fill up on, and I don't force her to eat the beans. If you put some good food in front of them and they throw it, then simply take the food away and mealtime is over. If you give in to their cravings for highly processed carbs now, they will only get worse and more picky as they get older.
At this age, your sons are old enough to have soft-cooked meat, and the iron and nutrients will be great for them. My little ones loved, loved, loved shredded chicken or turkey. (Dark meat has more iron in it than white meat.) Any meat that comes out of the crock pot is great. Fruits are good, in moderation. You'll know from their diapers when they get too much! The recommendation to avoid dairy until one year is for milk, not processed dairy products, so cheese, yogurt and cottage cheese are all great. Cottage cheese is an especially nutrient rich food. My daughters wouldn't eat is straight, so we mixed it up with some pureed fruit or yogurt. Beans are also a wonderful food - packed with iron and fiber. My second daughter loves them with just a bit of salt seasoning, or mixed with some pico de gallo style salsa. As your kids get older, a veggie tray before dinner is a great way to get them to eat vegetables. My toddler is so hungry that last half hour while I'm cooking dinner, that she will eat anything!
The most important thing to remember is that it is not your job to make your sons eat; it is your job to provide healthy choices for them to pick from. If you give in now under the desperate assumption that they are going to starve, you will just be feeding their pickiness (no pun intended).
Best of luck,
S.