“Can a generic USB network adapter be used, or is the Xbox 360 Wireless Network Adapter required to connect the Xbox 360 to a wireless network?”
The short answer is you need the Xbox 360 Wireless Network Adapter to connect your Xbox 360 to your wireless network. None of the generic USB network adapters work with the Xbox 360. There are a couple of other options for getting connected without re-wiring your house or buying the official Xbox 360 wireless adapter by using some alternative hardware.
A second option is to get a network over power line adapter, which allows you to put a wired connection close to your Xbox 360, pass the network signal over the wiring in your house and out to the rest of your network. Here’s a video on how this type of setup might work using the Corinex GameNet power line adapters.
There’s a ton of Ethernet Powerline solutions on eBay that are both affordable and effective for this type of alternative.
For more on Xbox 360 wireless network configuration, see this article on segmenting your wireless network for Xbox 360 gaming.