Ramona, you and I have the same technique.

I place them in very cold water after boiling them. Usually put ice in the water too and I never have a problem.

I'm surprised you can still eat chicken after having that job! LOL

I use a small spoon works good for me

If you can find an electric egg cooker, it works like a charm. Uses a small amount of water and it pricks the end of egg before cooking. Peeling is a dream. You can cook 7 eggs at one time. Last Christmas a "Bee Buddy" gave each one of us in our bee one. Said she got them at Bed Bath and Beyond.
Love it...

It is always easier to peel if you start with cold salted water in a pan then add eggs . Bring to boil and turn down and cook 15 mins. Drain the water and add cold water. Change water about every two to three minutes. Until the eggs are ready to peel. The salt help for the shells to peel without lossing the egg whites.

Had that trouble recently then learned the secrets. older eggs (like a week or two), very salted water, start them in cold, cook, then put in ice water to cool enough to handle, and they work great.

I think I'm going to go and boil some eggs

I never have problem peeling eggs since I dunk them in cold water immediately, usually even run the cold water over them. Maybe it shrinks the insides or causes the membrance to seperate from the shell.

My Mom told me this one, and it always works- put in 2 or 3 tbls of oil in when you boil them. Never fails.