Well. For me, that seems to have done the trick. I went back to newegg, to the motherboard page, linked me to the GIGABYTE product page, where there was a new LAN driver, posted just a month ago. I installed that and watched several videos with no problems. I think it is fixed. Hopefully this will work for you too.
Think about it. It just make sense - because it was video and file downloads, FLASH is not going to do it. Because the rest of my computers work fine, it is not the router, switch or modem. Because it happened on a new install with no other programs loaded, it was not a conflict with a virus program or something. I had the same problem, whether using Chrome, Firefox or Explorer. It just seemed to decide it had had enough data, and shut down - like the LAN didn't want to work. I should have thought of that a long time ago...