There is a dress code for Disney's signature gourmet restaurants. You can find the official dress code given in the question
HERE. The dress code for children is the same for children and adults.
There's really no need to be concerned about this -- even jeans and T's are fine in the signature restaurants. What you're going to want to avoid are cut-offs, sports gear, and swim attire. A rule of thumb you might want to think about for pants for the boys is that if the pants have a zipper and button, they're probably fine. If they're just pull-on shorts, sweats, or sport pants, then you'll want them to change.
Also, you should know that there is a signature restaurant that is in a theme park: the Hollywood Brown Derby at Disney's Hollywood Studios. I have never seen any sort of dress code enforced there. People just wear what they've been wearing to tour the parks. As long as you're not wearing anything offensive, you'll be able to dine there.
Have a great trip!