Is there a way to set Instagram as default camera app on iPhone?
As far as I know, the only way to do this is to jailbreak and use 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:

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.