What exactly is the grey wire in the back of the modem that appears to be a ethernet xable?
There are a LOT of different "modems" and "routers"; which model do you have?
That's a rather vague question. Maybe if you show us a picture we can tell you?
Husband says it goes to the box on the wall with all the cable wires in basement. I thought is was an ethernet, so I bought a Cat 6 wire because this "grey" wire is splayed open on the back of the modem. That's how it was left by last installer. Does it need to be replaced? Thought maybe this is the reason we keep having internet go out during the day.