This is not possible as mentioned. The reason for this is due to the MozyHome Backup Service located in services.msc requiring an active and functioning environment to operate, which is very reliant on the active Windows OS to exist. In terms of Mac OS X, the service in Activity Monitor requires an active environment to operate in as well.
Upon shutdown, multiple services begin halting and closing for both Mac OS X and Windows OS. This effectively halts Mozy from backing up immediately upon shutdown. Therefore there is no way to backup upon shutdown.