As the Big Red Button Kit listing puts it, “Arcade buttons are awesome, but they’re deep and it’s hard to find an enclosure that will fit them.” This kit, however, includes everything you need to add your own big red button. Besides the very large 60mm button itself, and quick-connect wiring, it also features a 3x3x3″ cardboard box on which to mount it. There is even a set of stickers to customize things.
If you do want to go further, you can 3D-print your own plastic enclosure based on your own ideas, or via this ready-made design on Thingiverse. Wiring feeds off to the side of the printed enclosure or cardboard, though with a bit of creative hot gluing, one could see an ATmega32U4 based dev board, or other HID capable device, stuffed into the housing itself. A few connections and a USB cable, and you can be in business with your own custom computer interface.
For a more rigid button/enclosure setup, featuring audio connections, be sure to check out the Super Smash Button that Chris D. featured last year!