First, to prevent eggshells from cracking, add a pinch of salt to the water before hard-boiling.
To shell the egg:
Gently tap both the top and bottom of the egg on your kitchen counter and run cold water over it.
Then roll the egg back and forth to crack and lift the shell away from the egg.
Run more cold water over the egg while peeling for extra help.
At this stage, you should be able to peel the entire shell easily.