Asking for my girlfriend, recently she purchased a used 4GB model for her brother, some games he is interested in playing require an official Microsoft branded hard drive, however he has asked me if it is possible to flash a USB drive for Xbox 360 in which certain games can also be compatible (for example, Battlefield 3, Battlefield 4), not just confiure a flash drive for Xbox 360 but also somehow make it unofficially Microsoft branded so these games can install on it. I personally would recommend purchasing an official 250GB hard drive but he does not want all of that extra space
You can configure a usb hard drive up to 2TB in size to use in the Xbox 360, you should be able to install games to that. You can also Hack a 250gb, 320gb or 500gb Sata harddrive to work in the Xbox 360 and Xbox 360 Slim.