Boil the quail eggs in boiling water (for 3 to 5 minutes or so). Put the boiled eggs in cold water.
Crack the egg shells against the sides of the pan, then make small cracks all over the surface while crushing the eggs with your fingernails. Peel off a bit of the rounded ends, and return the eggs to the cold water.
Peel starting from the small peeled bit you made to the bottoms of the eggs in Step 2.
The shell should come off in a thin strip.
I could peel these cleanly without blemishes or cuts.
Here's one that had been boiled for 3 minutes. It's cooked, but the yolk is still soft in the center.
Story Behind this Recipe
We use a lot of quail eggs in our osechi (New Year's feast food) so when I was peeling a bunch, I was able learn the trick to doing it in one swoop. You can peel chicken eggs in the same way if you make a lot of small cracks in the shell.
This method works by cracking the shells all over and introducing water between the thin membrane and the egg surface, If you can't peel an egg easily, just return it to the water and work on another one. I tend to work on the eggs in a clockwise direction, but if it's not going well I go counter-clockwise and that works sometimes .