MyBook Premium ES Edition

I have a MyBook Premium ES Edition, which is usb connectable. Can i network it though my Uverse 2wire wireless router with a usb to ethernet adapter? Do i need to update the software to have it network that way with both mac and PC?

There’s no updated available for that drive. Not certain it will work on a router trough Ethernet with an adapter. It will not hurt to try. Another option you have is upgrading to a My Book Live.