Most homes don't have a straight-up Ethernet connection coming into the house, but they do have telephone lines (as well as cable lines). It sounds like this is a phone lineâ€¦but I could be wrong.
That is to say, do you have a landline (analog) telephone service through AT&T as well, or is the U-verse service limited to television & Internet service (& digital telephone service)?
Also, is the router/U-verse modem the way you connect to the Internet? And which surveillance system are you thinking of using?