I'll be damned. I think I just found it. I turned off AVG and the problem did not reoccur.
Now to do more testing and maybe change anti-virus if necessary.
Edit: Just turned AVG on again and started the videos, ending them right after starting. Looked and was pegged out on downloading, even after closing FF. Turned off AVG while still pegged out and downloading stopped.
Tried the videos again with AVG off from the start and problem didn't happen again.
Edit: Information on AVG getting both FF and Chrome to do the same thing:
https://support.mozilla.org/en-US/questions/997796
Looks like AVG knows and is passing it on to the developers:
http://community.avg.com/windows_products?id=906b00000000LEsAAM