you may not need to reinforce the inside. I've had really good luck simply gluing box corners with Elmers.
I'd start by gently going over the outside with a stiff brush, then wiping it with a damp cloth and letting it dry. You can make temporary three-sided inside corner braces to hold things in place while the glue sets up. I make the braces out of priority mail boxes; just cut the corners out with a razor knife and make each leg about 2-3" long. you can put a little piece of wax paper between the inside corner and the brace, so they don't stick together.
You don't need much glue and it's best to work on one corner at a time. Dab it along each edge with a toothpick, use a small hand clamp or two to lock the corner brace in place, press the glued edges together and put a large rubber band around the perimeter to keep it tight.
This is my box repair kit. It's not fancy, but it works.
This High Standard box was a mess when I got it. The top and bottom were flat as a pancake (all corners separated) and one end flap was completely detached. It's not perfect, but it's usable and original to the gun.