She didn't simply started to care about him, he just suddenly become one of the three people she cares about most. It was pretty random and lazy from Lobdell to be honest.
Because Jason isn't a naive fool. He didn't declare his complete trust in JD and was pretty aware that she was shady. He just wanted to be wrong and tried to give her the benefit of the doubt, but quickly confronted her after his suspicion about her was confirmed.
His situation with his mother is a different matter and had a big impact on him in way that he shouldn't at all trust easily any person like he is strangely doing with Bizarro. He declared once that Roy is the one person he trust the most in the world, but it was never a blind one that we got one of the best sence in comics ever when he asked Roy if he was drinking again. It was just simple and great scene that showed the deep trust between them without making one of them look like a fool.