"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 Wireless Game Adapter, like the WGA600N from Linksys is one alternative to the official Xbox 360 wireless adapter. Using a game adapter like the WGA600N, you connect from the Ethernet port on the back of the Xbox 360 to the adapter. The adapter than connects to your wireless network. This has the advantage of dynamically switching between network frequencies assuming you have a compatible router to go with it, which can result in much better gaming performance. If your primary goal is simply to spend less money than the official Xbox 360 adapter, you won't save more than about $10.
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. I've done a ton of gaming using this kind of setup while gaming at Brandon's house. He's got more details on using a power line network adapter with your game console here.
For more on Xbox 360 wireless network configuration, see this article on segmenting your wireless network for Xbox 360 gaming.