There is absolutely no evidence at all to support the claim that there is any meaning behind which eye produces the first tear when you cry. This is a rumor perpetuated mostly on the internet.
Tears are produced in the lachrymal glands in both eyes. There's no preference based on emotion.
As a side note, when you cry, some of the tears in your eyes flow into your nasal duct, this is why your nose runs when you cry.