I'm not sure that's the mechanism at work because I have boiled eggs using plastic cups and also waxed paper cups many times. I don't think any water is boiling through. The cup itself is staying at 100C, which is below its ignition point.
The cup isn't exactly 100C, it's being heated by the flame. But it's only a few hundred microns thick, and the back is in direct contact with the 100C water. One side is exposed to an air/methane flame at around 2000C/3500F, which (while very hot) is a poor thermal conductor, and the other side is in contact with the water. The temperature at the interface is somewhere in the middle.
Also, you do you, but it's probably not a great idea to boil food in plastic cups or "waxed" (often coated in a styrene compound, not real "wax") paper cups...
Sure you have a 100C heatsink within millimeters, but I still find it surprising that the outside layer of the basket wouldn't burn away slowly.