Yes and no.
Officially, the only thing you can use on the 360 with XNA is C#. You can't ship any unmanaged DLLs with an app on xbox live, and if you want to write your game in C, you need a dev kit and an impossible-to-get contract with Redmond.
That said, anything you can compile into CIL that will run on the compact framework will be okay. If you can convince your code to compile as managed c++, you should be able to get it to work.
Times have changed. Grab VC++, your off-the -shelf XBONE and go nuts.
Or, better yet -if you're really stuck on C#, which is a pretty great little language - use Unity, or monogame. The former is a complete, really awesome engine and ecosystem, the latter is perfect if you want to roll your own.