One of the things that it took me years to find out how to do correctly is boiling eggs….. I mean boiling them without getting that green layer of color on the outside of the yolk. 
Now that it is important to me to have nice, bright yolks for salads and stuffed eggs, I’m glad I finally learned.
Here’s how I do it:
Put your eggs in one layer in a medium to large sized saucepan. Put in enough water to cover the eggs but don’t overload the pan. If the water is too close to the top of the pan and you think it might boil over, take some of the water or even some of the eggs out. Or try a bigger pan.
Set the pan on the burner and bring the water and eggs to a rolling boil. Immediately cover the pan and reduce the heat to a low simmer. Allow the eggs to simmer for just 12 minutes for large eggs ( 11 minutes if you are using medium-sized eggs and 13 minutes if you have extra large eggs).
Remove the eggs from heat, drain off the hot water.
Cover the eggs with cold water.
Shake the eggs around gently in the pan to break the shells, or break them by hand one at a time.
Drain off that water and add more cold water.
Allow them to sit in this cold water bath for another 5 minutes before draining and peeling.
Its true that the fresher the egg is the harder it is to peel once boiled. Something about air between the albumin and the shell. Peeled eggs should be used asap. Boiled eggs with the shell left on can be refrigerated for about 5 days according to the USDA. To help fresh eggs be easier to peel, add a pinch of baking soda to the boiling water.
How To Prepare Soft Boiled Eggs
I enjoy a soft boiled egg, toast and tea for breakfast. Not many people know how to soft boil an egg these days though. Most people fry the egg to the soft cooked stage. Here is how to soft boil an egg.
Bring your pan of water to a rolling boil. Add the eggs in the shell and cook for 5 minutes if you want a runny yolk and 7 minutes if you want a firmer, creamy yolk ( this is how I prepare them).
Serve in an egg cup or demitasse cup ( an espresso cup will do). Crack the shell with a tea spoon and eat the egg directly from the shell.
Hello and welcome! I'm Sylvia Britton, and I've been online since 1996 offering encouragement and help in the spirit of Titus 2 and Proverbs 31 at Christian Homekeeper. CHK is the original Homekeeping site on the internet
I boil eggs in the following way:
Begin as you say – putting eggs in sauce pan and cover the eggs with water. Place cover on pan. Bring the water to a boil. Turn off the burner. Set the timer for 20 minutes. At the end of this time the eggs are hard boiled and without that green coating you speak of.
I have found this works every time. Not keeping the burner on also helps save money.