Place eggs in a medium saucepan ; cover with cold water , bring to a boil ; reduce heat ; simmer about 15 minutes. Immediately drain , run cold water over eggs to stop cooking , peel eggs.
In measuring cup or coffee mug , combine 1/2 cup water and food color. Dip peeled egg ; one at a time into mixture for several seconds until desired color. Pat dry with a paper towel.
Cut thin slice from the wide bottom of the egg so it will stand straight. With a small sharp knife , cut off top of each egg , about 1/3 of the way down , either straight across or with slanted cuts to make a sawtooth edge. Carefully remove the top portion and yolk ; reserve tops. Place yolks in a small bowl.
Add mayonnaise and relish to yolks ; mash and mix thoroughly with fork. With a small spoon , refill large section of egg with the yolk mixture, heaping filling. Top each with reserved top portion of the egg. Use small specks of pimento's for the chicks eyes. Refrigerate 30 minutes or until chilled.