Thanks to everyone for all the advice! We do have character meals planned and will hope for the best. It sounds like every kid is different and we will have to just figure out what to do when we see their reactions!
I would second the advice to let them see characters in the parks before taking them to a character meal. If they refuse to go closer than 10 ft. or so in the parks, then it would be best to forget character meals, because there is no getting that much distance in a restaurant.
Some kids just never warm up to characters, but still love the parks. DS, now 8, has been to either DL or WDW over 20 times, starting when he was not quite 2. He is big enough to go on any ride in any of the parks, but there is no way on this earth he will voluntary get any closer than about 20 feet away from a "big-head" character. The one and only time we accidentally booked a character meal (didn't realize until we were seated) we tried to just keep them away from the table. Disaster. DS spent the entire meal underneath it, with his eyes closed and with a death grip on the table leg, repeatedly praying "Please, God, make them disappear!" He was four at the time, and every time they got close enough to have their feet become visible under the table, he would start shaking. We actually had to ask the handlers to have them all leave the room momentarily so that we could get him out of there without using force to pry him off the table. Nowadays, he just goes to great lengths to avoid them: a few months ago at BB, he walked halfway around the park in order to avoid the characters who were standing on a bridge between the location of our beach chairs and the entrance to a slide he wanted to go on.
Just FYI, character-avoidance has a great upside: we never spend half our park day waiting in autograph lines instead of going on rides, and we don't have to bother with the expense of character meals, either.