It is very possible to get infection from blocked tear duct. Usually tear moisturize our eyes and kill 95% bacteria in eyes. Since there are no blood vessels on cornea, it gets nutrition from tears. We could tell how important tear is in eyes. Yet normally tear is drained from the inner corner of eyes into nasal cavity through the internal passage. Once this passage is blocked, the tear builds up in eyes, which would cause eye infection and thick secretion. You can use some salt water to wash your eyes to reduce the green thick pus in eyes, which at the same time can control the infection of eyes too. Massage the inner corner of eyes near to nose bridge which could help break the blockage. Normally for adults, it is related to some sinus problems, or abnormal condition in nasal cavity or thickening of the tear duct lining. I suggest you to find out the underlying cause which can lead to optimal treatment.