The beta won't work from a flash drive -- you HAVE to have it on an HDD (which it doesn't tell you until you've fully downloaded it). So before downloading, move content from your HDD to your flash drive and then download the beta to your HDD.

What is it with Bungie games requiring an HDD? Every other game I've ever tried has worked perfectly fine from a flash drive.

And if you were planning on using XBL Gold trial membership, you're also out of luck. WTF?

http://www.bungie.net/en/Help/Article/11909

Error Code: BUZZARD
A PlayStation Plus or Xbox Live Gold account is required to play the Destiny Beta. You can acquire a PlayStation Plus or Xbox Live Gold membership from the PlayStation Store and Xbox Marketplace, respectively, or from participating retailers.

While Xbox Live Gold trials may work on Xbox One, they will not work to download and play the Destiny Beta on Xbox 360

If you are an Xbox 360 player you must have a full and active Xbox Live membership to participate.