It's basic mechanical engineering and the same reason arches and domes don't collapse, and the reason why corrugated sheets can support more weight than flat ones.

It WILL collapse if you press hard enough. The only reason you are surprised is because you are thinking in terms of the load-bearing capability of a flat sheet of eggshell-like material. The basic reason is that the stress/strain behavior of materials depends on shape as well as intrinsic properties. The shape helps use the strength along one dimension to support a load along another while preventing buckling.

You do basic stress/strain calculations on things like this in undergrad mechanical engineering. The shape is a little complex, but a perfect sphere will behave the same way.

If you want some non-mathematical intuition about why this works, look at an arch or dome. There you have discrete wedge shaped pieces where downward forces cause inter-wedge normal forces, which have a component that counteracts the stress. The harder you push, the tighter the pieces get wedged, and the stronger the reaction. The smaller the radius of curvature, the stronger the shell (in the arch analogy, the wedge pieces narrow more, so the component that pushes back is stronger). This is why you can crush the egg the other way more easily (horizontally).

When you push hard enough that a break occurs, it will normally start due to some non-uniformity in a large-radius part, like a tiny crack.

But really, this is far easier to understand mathematically. Some things just aren't meant to be understood through words. Look up "thin shells" in any mechanical engineering strength of materials type textbook.