Step 4. Turn off the heat and cover the pot. Allow the chicken legs to cook in the hot water for roughly 15 minutes, until the chicken is cooked to an internal temperature of 165 degrees Fahrenheit for safe consumption.hen you're boiling the chicken legs, add seasonings to the water to enhance flavor -- the resulting broth can be saved to cook other dishes, including chicken soup, later. Remove the skin from the chicken leg quarters and season with salt and pepper to taste. Place the legs in a large pot.