Four Points Sheraton does, but you really don't have a long enough connection to leave LAX (depends on which airline to Boston and day of week). You'll need to leave the hotel by 11:30 at the latest to check-in for a 1:30 flight (check with the airline for the LAX wait times, the worst I've seen was 3 hours but 30 minutes is more common). Clearing INS and customs will take an hour, so you probably won't get to the hotel before 9. If you belong to Qantas Club you can use the Admirals Club in Terminal 4 (this has showers available). Check what lounge access you might have available.