Same problems with Cox and my AEBS that just popped up over the last week or so. Cox tech support makes you take the AEBS out of the loop and then when you can connect directly to the modem tells you the problem is on your end (and it is, kind of).
My AEBS was set (by default I believe) to configure IPv6 automatically in "Native" mode.
This resulted in the AEBS dropping the internet connection daily and sometimes several times a day. Same symptoms as the OP and the only temporary fix was to restart the AEBS via Airport Utility or unplug the AEBS from the wall and restart it that way. This fix would last for no more than 12-24 hours, IE I would go to bed and the connection was dropped by the morning.
I found this thread and changed the IPv6 configuration to "Link-Local' and so far all is well. I will test some more over the next few days and report back.
If I had to guess Cox is rolling out IPv6 and there is some type of incompatibility with their protocol and what the AEBS uses.
My neighbor across the street is having the same issues with almost exactly the same equipment I have and Cox also. In fact he asked me a week or so ago if I was having to reset my router too. This seemed to start the day before Thanksgiving, 11/25, in my area (Baton Rouge, LA).