Is there a verb for online-ness and offline-ness?
Connected and disconnected.
As far as I know, to come online and to go offline are used in different online chat rooms/MMOs' ingame chats and other such systems.
Edit: Your own suggestion is another good variant, applicable if there is someone/something else that performs the operation. E.g. My friend went offline right in the middle of discussion vs. We had to take the server offline to perform maintenance on it.