No. The TV signals present at the ethernet ports on the U-verse gateway (modem-router) cannot be received by any means other than a U-verse receiver or DVR.
Also, there is no "basic cable" on U-verse. TV channels are available only in their designated digital packages. They all include the local channels generally considered as "basic" service, but again, you can't get them without a U-verse receiver.
There is really only one way to add a TV to a U-verse system without getting another receiver: connect it to an existing receiver with a long cable. You could use the coaxial connector on the back of the receiver (looks like an antenna jack), and run the coax almost any distance in the house. The downside of that is that the received signal will be analog--HD isn't possible that way.
You could make it HD with a long HDMI cable instead of coax, but there are practical limits to the length of such cables.
Also, a major downside to either of those: channels can only be selected on the U-verse receiver. Both TVs will always see the same channel.
Have you considered simply getting a good antenna for the 2nd TV? Great HD, and no U-verse or cable costs at all. (But it needs to be a *good* antenna.)