From Joomla! Documentation
"Note that jdoc:include elements are not parsed in these error pages."
This can´t be right - I have a offline.php override in the root of my template. And in this I inserted a module Position which really shows up. So it would be possible to make additional Positions just for the Offlinepage (eg offline1, offline2, etc...) or even put some existing Positions in it. May the Newsfeed-Modul or something similar.
I have deleted the "Login Form" from the offline.php and inserted a Module-Position "login" instead. Now I can control via the Module Parameters the behaviour of the Login-Modul on the Offline-Site. By this way every Login Modul can be used, doesn´t matter if its the Core Login, Community Builder Login, SignalLogin or whatever, so there is also no stop for absolutly restyling it.
But i have to say: The only Login-Solution which aint work is the LoginBox. I think the component needs the System enabled to show the Loginform in the Lightbox, but I´m not sure.
May somebody know it?