I think this is because when you push on the sides of the egg you are pushing the shell in towards the center over a relatively small surface area it is not very strong. When you push on the ends of the egg, a lot of the force you exert is actually pushing along the surface of the egg and it would be stronger in that direction as the force is more distributed.

Apparently, if you place an egg in the palm of your hand and close your hand so that your fingers are completely wrapped around the egg, then squeeze it, the egg will not break. You should try it yourself. If you are worried about making a big mess, just do it over the kitchen or lab sink. (Make sure you are not wearing a ring, though).

Eggs are amazingly strong despite their reputation for being so fragile. Eggs are similar in shape to a 3-dimensional arch, which is one of the strongest architectural forms. The curved form of the shell distributes pressure evenly all over the shell rather than concentrating it at any one point. While you squeeze the egg, you are applying even pressure all around the shell. But, eggs do NOT stand up well to uneven forces which is why they crack easily on the side of a bowl, or if you do the same experiment while wearing a ring. To check out this experiment you can go to this website:
squeeze an egg.