J.,
I agree it sounds like an eye infection + a blocked tear duct. Essentially the drainage canal (near the "ball" of the interior of the eye close to the nose) is blocked at birth. Sometimes it can be "mucus" that's stuck, othertimes they think it's a thin membrane that never opened properly.
Most of the time you'd never know it. But when they cry you see tears only drip out. When it gets infected you notice it's glassy, and later becomes gooey. I noticed it more on my kids once they started solid foods because they were always rubbing their eyes with dirty, food covered hands.
The infection does need to be treated with antibiotics to prevent scarring of the eye and further discomfort. If you get past the infection, the only way to "unblock" the duct is to massage the side of the nose, starting close to the eye and rubbing downward along the side toward the nostril.
If you don't get it opened by the time the child is 1 yr old, peds/ped opthamologist recommend a quick surgery to "pierce" the duct open to prevent future infections and scarring. My second child did have to get the surgery. It was very quick and had no complications. Out of 3 kids, we've had to deal with 4 blocked tear ducts. Two resolved themselves, one needed surgery and one is TBD.
Good luck! Email me with any questions.
Sara