That is not the quote...It reads more like...if he didn’t do anything, it would bother him....it’s called conscience
Since the prevailing theology of the time (and even our time) is that children are born with a sinful nature, I think it entirely possible that Lincoln was taking the opposite side, but if so, I can't see the preacher arguing that Lincoln's decision to save the piglets was motivated from an evil nature.
The story, I think, is much like the C.S. Lewis conundrum cited in the post.