Why cant we cook Eggs in Microwave?

We can cook and egg in the microwave. You have to cover the eggs by at least 1 inch of water and when you have that much water the microwave oven becomes quite inefficient; it'll take about the same time as on the stove top. The only advantage of boiling eggs in the shell by microwave is that the oven turns off automatically after the cooking time has expired.

You can, just not in the shell. Microwave ovens work by causing water molecules to vibrate, which causes heat--and steam. When water becomes steam it expands and needs to escape. Think of putting a covered pot of water on a stove and heating it to a boil; the top will rattle as the steam escapes. The egg shell has no way for the steam to escape, so it basically explodes when the steam builds up.