Sorry I have to correct you here. The squeeze gesture is picked up by 2 rows of sensors inside the edges of the frame. It has nothing to do with the screen.Someone will surely correct me if I'm wrong, but I don't think you actually squeeze the body of the phone, meaning, the sides don't actually compress. What you're doing by squeezing the phone is really squeezing your fingertips (and the base of your palm area), so that your skin makes contact with more of the extreme edges of the screen. It's very slight, but enough to activate the sensitive touch areas there. When the phone sees a whole bunch of "touch" along the sides of the screen, it sees that as a squeeze. So, you don't have to worry about a case cracking under the pressure. However, if this really is how it works, then any case with screen-overlap may interfere with activation of squeeze command. Ditto with screen protectors.
As for the P in P-OLED. That refers to the substrate, which is the base to which all of the electronic bits are connected. Plastic allows this substrate to be thinner and more flexible. The outer surface may be, and often is, still made of glass (in this case, Gorilla Glass 5). P-OLED isn't typically used to make the body of the phone more damage resistant (although it certainly can be). It's used to make the screen a tiny bit lighter and thinner, allowing more room for other internal pieces (such as a slightly larger battery).