Will the airport's X-ray scanner damage my Xbox 360?
You'll be fine. Millions of passengers have no problem with laptops brought through airport security. Your XBox will be fine.
Here's a quote from About.com:
Letting your laptop go through the x-ray equipment will not harm your laptop. The magnetic field which is generated is not enough to cause harm to your hard drive or cause damage to your data. Metal detectors on the other hand, can cause damage and request politely that Security/Customs do not use the metal detector but do a hand check instead.