Can I change the default camera app on my iPhone?

There is no native way to do that.

As far as I know, the only way to do this is via a Cydia app called AnyLockApp. It allows you to configure the behavior of the camera button that appears on the lock screen. You can set any app to be activated by that button. See also the AnyLockApp page.

Settings screen:

Edit: As mentioned in the comments and by stuffe's answer, this hack will probably not work in iOS 5.1. However, this is unconfirmed. I will update this answer when I do update and jailbreak iOS 5.1.

Besides this hack, there is no other way to change the behavior of the camera button on the lock screen.

Other than jailbreak you are out of luck, the jailbreak version detail in another answer will likely work on 5.0.1 (double tap home to reveal the camera button and volume controls) but in 5.1 it's not a button any more, but a sliding control and as such highly doubtful that a jailbreak hack will be able to modify it's function.