My first suggestion would be to try to pack your food either in disposable bags (lunch sacks or grocery bags) or a collapsible/foldable cooler. Otherwise you end up having to lug around an empty bag.
We frequently pack lunch or dinner to take on flights. Be careful of a few things.... number one is foods that may cause allergy issues in others like nuts (peanuts and tree nuts). Sure, there is ALWAYS something that someone is allergic to, but peanuts and tree nuts seem to be the "biggies". Second, be aware not to pack anything too "stinky", because you will be consuming it in a fairly confined space. I would avoid tuna salad and egg salad because they are both rather smelly.
Sandwiches made on bagels hold up well without getting smushed. Pretzels hold up well. Pack them in a disposable container like an old Cool-Whip container or a small cereal box you can tape shut and they won't get crushed.
My kids enjoy lots of little treats instead of a regular lunch. Crackers and cheese, fruit (raisins,grapes, bananas, apples, blueberries, strawberries) that doesn't require a knife to eat, cheese sticks, dry cereal in baggies, guacamole (100 calorie packs are easy) with crackers, baked chips, or veggie sticks.
It is hard to tell you exactly what to pack because they have tightened up the "carry on" rules so much. I wouldn't risk taking things that I always used to take like yogurt tubes, pudding tubes, and other "semi-liquid" things like that. Not sure what the rules are about things like fruits. You may have to check the FAA website for specifics. You will DEFINITELY need to buy drinks after you get through security. Or bring empty water bottles to refill at a water fountain. Or the Subway idea is starting to sound pretty good

.... Good luck regardless of what you decide to do. Have a great trip..................P