This is getting serious mama, time to change your plan of attack. It sounds like this is turning into a battle of the wills. You don't say how old your children are...I am guessing school age? I think the first thing I would do is get everything they are wanting to eat out of the house. Hide it, stash it or trash it. Whatever you need to do. Do it while they are sleeping, so they don't know it's happening. If you offer up applesauce and they see a Squeezer (??not even sure what that is ;) ), then of course they are going to beg for it and refuse the apple sauce. Get the fancy packaging out of the house.
I would first have a serious meeting with the family. At the dinner table. Explain to them that all food will be eaten at the table. And don't give in. Even if they are eating something that's easy to clean up or you just don't want the arguement. Explain that you expect everyone to sit at the table at meal time. You must be there too. Show by example. And let them know that you expect them to sit with good manners. "Yuck." "That's Gross." and other mean and impolite comments are no longer allowed. If you don't like what is cooked, you don't have to eat, but you need to sit here with us and talk. Start some sort of tradition to keep them at the table even if they don't want to eat. Ask them their favorite part of the day. Ask them to tell you one bad thing about the day and then three terrific things. Count , ABCs...whatever it takes to keep them at the table. The longer they sit with you, the more likely they are to eat. Whether it's cleaning their plate or nibbling.
Get them excited about food. Let them help make the menu. You'll have to compromise on this, but give them some input. Have them help you shop. Start a garden. Grow some herbs they can pick for you. Take them to a farm. Anything to get them excited about the idea of good for you foods.
Some kids are just picky. It happens. I was one of those kids. You could have shoved broccoli on my plate every day for years and it would have made me gag every time. So after years of arguing, my mother got smart and learned to work around my picky behaviours. Obviously they eat something...time to build off that. Plan your meals around the few things they like. Are they fans of pizza? Sneak some good stuff in the sauce. I have a friend who was shocked her some was eating our "white pizza" because it had spinach in the sauce. The kid had no idea and happily ate it. Then when he was going in for the third slice, my son spilled the beans and the kid suddenly thought it was "yucky". You can hide a lot in spaghetti sauce. :) Do they like chicken nuggets? Fix that with a healthier veggie. They can "earn" extra nuggets by taking bites of other foods.
Sometimes presentation is everything. I know lots of people that can get their kids to eat things by letting them dip it. Try yogurt to dip the fruit in. Make fondue in a crock pot and let them dip things in cheese. Sometimes I am truly amazed what my kids will eat if I let them use toothpicks instead of forks.
Try getting them in the kitchen to help. First they will love it. And most kids who cook are excited about the food they made. Make a big production about them cooking dinner. MY daughter is 2 and even she helps me cook. Our favorite "kid cooking" dinner is simple. We cook up pasta, they choose the shape. Then they help me roll turkey into small meatballs. We just add a little bread crumb and some italian seasoning to the meat. I buy mozarella balls that I let them cut in half. This can be done with a butter knife. We add cherry tomatoes to it and sometimes cooked veggies from our garden. Top it with a little parm. It can be as simple as letting them crack an egg for you.
I also would not do dessert regularly. One night when you notice them eating or behaving well at the table. That's when you pull out a dessert. "You guys did such a great job, let's celebrate!" My kids never know if there will be dessert and it didn't take long for them to realize that it doesn't happen often when you ask for it...so they rarely ask.
Hang in there mama. Stand your ground on eating healthier, but compromise a bit too. Good luck!