Will pink eye make a tear duct swollen?

Dacryocystitis: Obstruction of the nasolacrimal duct, with or without a dacryolith, can lead to dacryocystitis, an infection of the tear duct into the nose. Actinomycetes species are a common culprit. The infection can spread to the conjunctiva and cause a chronic "pink eye." an ophthalmologist can probe the duct, which might be curative along with antibiotic therapy. A dacryocystorhinostomy is sometimes needed.

Possible connection: Yes, infact it may be the other way around. A nasolacrimal duct infection can cause pink eye especially in the medial angle(towards the nose).