Ask your vet to perform what's called a Schirmer Tear Test. When dogs don't produce enough tears (most likely due to a blockage or defect in the tear duct) they'll produce this grey, stringy mucus. If there's any redness or he seems to be pawing at his eyes definitely take him in as he could have any number of issues like conjunctivitis (which is zoonotic, meaning it can pass from him to you).

a vet should examine your dog for problems with his/her eye lids.

this link is more about removing tear stains but also offers some help in your case. http://www.bukisa.com/articles/18794_how-to-remove-tear-stains-in-dogs

My dog was doing the same thing only it was a white matter in his eyes. They were also red. I took him to the vet and he had conjuntivitus. They ran all kinds of tests on him. I have to give him medicine from a tube and also eye drops 2 times a day. After 3 weeks of this his eyes are looking great. He will have to stay on this medicine for the rest of his life. Hope this is helpful.

Mine seem to have the same problem and Mojo only started getting goopy eyes in the past year. I had a vet tell me it was allergies, in the meantime I spend a lot of time chasing them around with tissues.