I"ve always heard "break my heart FOR what breaks yours", but actually both could work. We want our hearts to break and take action for the things that break God's heart, and we also want our own hearts to break away from the sin that breaks His heart.
As far as Biblical basis, I'd say any sort of verse pertaining that we are to be like Jesus, or to seek after God. The more we strive to be like Him, the more the material things of the world will fade and the more our hearts will be tender towards those who are hurting or are sinning.