Answer
When an egg is squeezed along the long axis, the arches of the ends are symmetric and shaped like catenaries. Pressing the ends together applies symmetric compressive forces that are tangent to the eggshell. When squeezing from the sides, this is not true, and the eggshell bows outward and breaks.
Try it yourself, over a sink, and wearing gloves to protect your hand from eggshell fragments.
This is discussed on page 234 and on page 236 in the Check Point.
