YES!!!! Enda's weep terrible just going out places, I have a rag I carry around the outside of the show ring to wipe to goobers out of the corners of her eyes before we go in. Usually if it's just a casual visit to the conformation ring it's just a little goop that keeps building, but they leak quite a bit when she's stressed, to the point where down either side of her face is a wet river. After noticing the goop enough times when we would go to show I just pinned it down to nervousness, and if something happened that made her otherwise noticeably uncomfortable is when I started to see her eye actually run, so I do believe that that is one of the signs of stress in a dog, runny eyes.
Also her eyes run quite a bit just around the house on really hot days after she's been worked or playing, again a form of "stress", although she's happy and satisfied her body was stressed.