When we want to boil eggs we wash them and put them in the refrigerator for a week or so. Then my DW sticks the end with a pin before boiling. She claims that makes them peel easily. We put the eggs in a pot, cover them with cold water and a tsp of salt, boil for 8 minutes, drain and cover them with cold water and ice for at least 8 min. All I know is we did 2 dozen last week that were laid at the end of October and everything peeled easily.
BTW, we had one of those egg peelers--was a bigger mess than doing it by hand--think it went in a yard sale or was sent to the landfill.