Single Perfect Tear
This is sometimes referred to as the “Single Emo Tear of Manly Angst” (which originated on TWOP) or variations thereof. The term also spread to the writers' room, as evidenced by this quote from Sera Gamble:
Abandon All Hope... was the first time I actually cried watching a director’s cut. At first I thought maybe I was just having one of those days, but I checked in with the other writers… I don’t want to incriminate anybody, so I will just say that one or more male producers also shed what we like to refer to as A Single Perfect Man-Tear.
This tear was referenced in 10.05 Fan Fiction in Supernatural: The Musical!, with the song "A Single Man Tear", sung by the actors portraying Sam and Dean for the musical, with "Sam" reflecting that Dean hides his emotions, afraid of how others see him, while "Dean" acknowledges his repression of his feelings out of fear of his inability to forge his own path.
It is unknown if this ability is in anyway related to Jensen's Wonky Eye.