Thanks to both of you: the tests you suggested confirmed that it's a hardware issue. I plugged the NAS drive into the ethernet port on one of our PCs, and the connection light on the drive came on. When I did the loopback test on the router, using the same cable that I'd used for the NAS drive test, the ethernet port lights did not come on.
I'll contact Verizon to have them swap out the router. Thanks again!