U-Verse is supposed to be "fiber" everywhere. But in reality, it is a weird hybrid of fiber to the node (usually green boxes you see in some people's front lawn) and copper (phone line) to the home.
Ideally, every U-Verse customer would have fiber running to their home/building, but that is very, very expensive. I know in some cities, AT&T does do that, but I don't think they do in most cities.
AT&T U-Verse is a replacement of AT&T DSL where available.
So you have a VOIP system. Your phone goes through the AT&T gateway. They refer to this as "Digital phone service" or something along those lines.
Does your Internet also go down when your phone service stops? I also have AT&T U-Verse and fortunately haven't really had these issues other than the fact that it isn't as fast as I'd like.
The thing is that if AT&T is installing U-Verse in your area, they probably won't have AT&T DSL anymore, so you'd need to find a DSL provider if one even exists in your area. If you get DSL, AND your phone service isn't digital, then you will simply plug in your phone to a wall outlet. Of course, you will need a DSL filter so that your DSL modem and phone will work properly.