Sounds like his sinuses were collecting infection. When the sinuses or ears are infected they pus has to find a way out. It comes out the eyes, which your little ones did (It's not pink eye, it's pus from inside, not from being exposed to a disease), and it comes out the ears, the nose, down the throat, any way it can find a crack to get out it will.
Get a sinus syringe that is like a balloon on one end. It has a long thin thing on the other end that goes into the nose.
Wrap the baby up in a blanket, hold the baby's head in your left forearm and pick up the syringe in your right hand. Use your left hand to hold the nasal opening closed on the nostril you're not suctioning out. This way it sucks from inside and not from the other nostril.
You can squish the goo out onto a tissue then go back in for more. Once you're not getting gunk out do the other side.
This is what kept my kiddo's from having super sinus and ear infections for most of their young lives. The mucus sits and grows bacteria then it starts draining out. It's nasty and the kids hate it but if you do it a few times per day when the nose is gross it will keep you and your kiddo healthier.