Baby shampoo is the best! The trick is to keep using it twice a day until you are sure the pink eye is gone, usually about 3-4 days. I just pour it on my fingertips, lather them up with a bit of water and rub the closed eyes with the shampoo. I always try to open my eyes just a teeny bit so the shampoo can get in there. It doesn't hurt or burn at all. Afterward, rinse well and dry. Sometimes I follow it with coconut oil (usually only at night though) rubbed onto my closed eyes.
When my daughter was young I read in a home remedy book to mix 1 part baby shampoo to 10 parts warm water. Use a cotton ball to clean outside of eye to remove mucus from lashes, then numerous times each day wash eye with same solution, warm is easier on the eye. It worked on my daughter each time. We now use it for the grandchildren when the need arises.