Can't time capsule back-up to an external hard-drive?
I used Apple's Express Lane tech support. (Being as this was a brand new TimeCapsule, there was no fee.) 

The tech told me that TC no longer supports backup to an external drive as of the 2nd Gen TC. He would not speculate as to why.

I can speculate that perhaps Apple wanted you to buy a bigger TC instead? Or perhaps (as was my experience) backing up to an external drive seemed to cause the TC to shut down (maybe overheated).

However, I can and did hook the drive directly to my USB port and I can back up that computer to it using Time Machine software (which is what I wanted to do anyway.) 

Odd to answer my own question but this thread may provide an answer to the next guy.