The standing rule in our home is and always has been:
You try whatever you are served. If you don't like it, you are free to make yourself a sandwich.
My kids are very non-picky eaters because of this. I honestly can't keep fruits and vegetables stocked in my house because they LOVE them - but I know that is on the rare side. They also have tried (and loved) things like frogs, sharks, alligator (thank you cruise ship) and every kind of seafood my husband has ever had. I think they do this because of our rule from the time they were babies. They don't have to like it, but they have to try it. It's rude in my eyes to not try what they are served. We don't offer them liver and onions, so they can eat at least one bite. MOST of the time they end up loving it.
Maybe try that rule in your house, but start slowly since they are older? Don't go too outside of the box.
Also, if I know my kids don't like something but it's not horrible, they will eat their age in bites. At least they used to. At 7, 9, and 11, they know to eat enough to make me happy. They have also always thrived on knowing WHY a certain thing is good for them. Protein keeps them full longer (meats, peaunt butter, cheese, etc), carbs give them energy (pasta, potatoes, etc)....
We always take a huge bag of fresh vegetables on any picnic we do. It always has carrots, peppers (normally orange and yellow), cucumbers, broccoli, and celery. We pack little things of ranch and peanut butter to go with them. One time I didn't have any peppers so I just omitted them. My then 5 year old asked where the vitamin A was!! Ha!! He knows we buy the bright colored ones for their increased vitamin A amounts and wanted his fill :). Make it fun for your kids...teach them about the food, how it makes them strong and smart, what specifically it does for them...I bet things turn around in time.