Wrong logic. Inverse logic.
Why would the air in the lungs compress? The air in my scuba tank does not compress during a dive. I can clearly see how the SPG shows a constant pressure even when I dive from 1 ATM to 2 ATM. The pressure drops with time but depth does not affect cylinder pressure. This has something to do with the STEEL cylinders.
The air in my lungs would then retain its pressure. PV/T=constant and as my body temp is contant, so PV=contant. Volume goes down, pressure up, or vice versa. At depth there will be an external overpressure (caused by the weight of the water) that compresses the body - and hence the air inside. When the volume has shrunk and pressure increased accordingly (to match the external pressure) an equilibrium is reached.
And oh, it's not my rib cage but my belly that is giving way. The soft tissues.
(I have tried to compress my rib cage during some tight crawling and no, it's not that compressible.)
Scuba diving is different because then we breathe compressed air at the ambient pressure.