Just an FYI: The reviews of the Piaggio 350 tail box and mount were not that great. I decided to go the Givi route. On Modernvespa.com there was a person from the UK who has a BV350 who stated that the Givi rear rack mount for the BV300 is the same for the 350. I could not find a US supplier anywhere who listed what I wanted.
The Givi part number is E349. You remove the little plastic top piece on the rear rack of the BV350 and the Givi piece bolts into place. You can then use any Givi MonoLock (MonoLock, not MonoKey) top box.
I had to look for a while but finally found a place in Europe who would ship to the US and ordered the Givi E349 piece. Shipping was expensive and the who part + shipping came out to almost $90 USD.
I already have a 30 litre Givi Monolock box so I just needed the mount. This just a FYI for anyone in the US looking for a rear top box who doesn't want to use the Piaggio one.
I got my part at this link. This is just the mounting plate.
Here's a shot of the part and installation instructions from Givi. This was a PDF I found somewhere and downloaded.