I have used breastmilk on plugged eye ducts, baby acne, diaper rashes, you name it, and usually within 2 days it begins to improve. IF it gets worse its best to see a doc. I never tried it on pink eye though.
Poor baby, my daughter had something like this but it was a cold in her eye, it wasn't red - and I'd squirt her in the eye while she was eating, I know it sounds bad, then massage the tear duct to make the eye water and clean itself, then I'd squirt again. I always used straight from the breast as there was less of a chance for contamination via bottle or dropper.
If it doesn't improve over 3 days Or gets worse deff. see a doc as breast milk may not be doing the trick.