If you want to play or downloaded game files (e.g., from your own discs), you need a modded console or an RGH/JTAG hacked Xbox 360. This method bypasses the 32 GB limit and allows external drives up to 2 TB.
You must own the physical disc to install the game (and it must remain in the tray to play). Step 1: Prepare the USB Drive install games xbox 360 usb
Microsoft officially supports using USB drives for game storage. Here’s how to set it up: If you want to play or downloaded game files (e