Requires iOS 7
Requires Paper for Facebook app available from App Store.
For iOS 6, see Message Box (iOS 6)
Set your Facebook Chat Heads free!
The Facebook app has Chat Heads locked down to being in-app only, this'll help fix that! Take them out of the Facebook app and do whatever you'd like!
A manual chat refresh timer is configurable in settings, though keep in mind a shorter delay will decrease battery life.
*Note* It's much easier if you have the Facebook Messenger app either logged-out, or uninstalled. That way you won't get confused with the push notifications.
Source code available on Github
Configure options from Settings app.
See larger Screenshots below..