On all the bikes I've had the DRIVE SIDE has tightened up ANTI-CLOCKWISE (ie lefty-tighty).
The NON-DRIVE-SIDE has tightened up CLOCKWISE (ie righty-tighty, aka the "normal" way).
you said you unscrewed the non-drive side and the bottom bracket popped out?
Usually with the Shimano sealed bottom brackets the main body is screwed into the BB shell from the drive side, and the plastic or metal "sleeve" is screwed in from the non-drie side.
Maybe it had been installed incorrectly?
What frame is it in?
This. AFAIK there are exceptions - UN70 and some from the early lower LP range had the removable plastic cup on the DS. Italian is threaded clockwise on the DS - which this most certainly is not and it would be difficult to bodge since the diameter is larger than BSA.
Favorite quote on Retrobike:
Not worth the petrol to take it to the tip so it's down to the angle grinder to make it small enough to put in the wheely bin.