If this were me, I would make sure auditing is turned on and then I would audit the connect/disconnects. This would give me a database perspective of what is going on during all times.

Also, I would create a DDL trigger that would capture the shutdown/startup of the database so I have proof of when this beast was down.

I would also write a script that tried to connect to the database every X minutes and record when I couldn't connect.

Oh wait, I already do this, forget it.