Short answer
Eggs can be microwaved after being removed from the shell, but intact shell eggs should never go into a microwave. Yolks and already-cooked eggs can also trap steam and burst.
Why explosions happen
Microwaves heat water within the egg. A shell, yolk membrane or firm cooked white can prevent steam escaping. Pressure may build until the egg bursts, sometimes after the oven stops when the surface is disturbed.
Safer microwave cooking
Use a microwave-safe container, remove the shell and break or pierce the yolk. Cover loosely rather than sealing. Use lower power or short intervals, stir scrambled preparations between bursts and allow standing time for heat to equalise.
Reheating cooked eggs
Do not reheat a whole peeled hard-boiled egg intact. Cut it into pieces and heat gently as part of another dish if appropriate. Microwave power and portion size vary greatly, so rely on even heating rather than a single universal time.