Destiny won't use external storage for whatever reason, only the hard drive. You can put other games' stuff on the external storage though (unless they also refuse to use it, but I think only a handful of games do that). I haven't used external storage for games much but I can't remember hearing about anyone having issues with it.

I dumped all my freebie Games on Demand games on my flash drive. I've never had a single problem running any of them, and load time seems to be on par with having the disc installed.

Destiny only runs on the hard drive. I just moved everything else to flash drives, and it is fine. All the games work great on those. Remember, they just have to beat DVD read speed.

Others have mentioned that Destiny won't run off of flash drives, but even if that weren't the case I'd still say just get a bigger hard drive. I'm seeing the 120 and 250GB versions going for between 30-45 bucks on eBay right now. All you need is that and a transfer cable (which I'm seeing for 5-10 bucks).

Once you get your new, bigger drive, you plug it into the console with the transfer cable (leaving the old hard drive attached where it currently is). The 360's OS should detect that there's a transfer cable in use and ask you if you want to move things to the new drive. Wait for it to complete, power down, and swap the new drive on to the console. Boom.

Be advised that the process MOVES data rather than copying it. So once it's done, the old drive won't have anything on it. If you've only got the one console, that probably doesn't matter, but just so you're aware.

I think I have an old side mount 60gb hard drive you can have if that's the correct style of HD you need