Aniother thing I would do when carrying on a box (or checking for that matter):

I always have a 60" or 40" webbing strap, 1" width and fully adjustable with side-release plastic buckles in my carry-on. They can be really handy. If it's a small box I can rig the strap for carrying it like a bag so I don't have to use both hands. (And if for some reason my accessory strap fails or is lost I can fashion a temp replacement). REI has a pretty good selection of webbing straps.

In the event screeners want to open the box, I'll usually have a small roll of packaging tape with me to reseal it. Sometimes tape is readily handy for a TSA screener to reseal, sometimes not. You can save time by bringing your own.