All the pros and cons aside, yes, you can run Blizzard games from whatever drive you want.
To do that, just move it to the USB drive. The launcher on the other hand resides on the PC playing the game. Just point the launcher at whatever drive has the game files. Remember to patch the game as well before playing. That does involve pointing the launcher at those files and telling it to update unless you have it set up to auto update the correct location.
The only thing you might have a bit of an issue with is the D3prefs.txt that handles your game settings for graphics and a few other things. If you have any custom settings, then you may want to write those down or keep a backup copy of the file. That is located in your Documents, not program files.