I don't think it's so much that she starts caring when her boyfriend is in danger so much as she starts allowing herself to care in her relationship with Zuko before that. Their relationship and his example is I think the thing that starts to crack through her whole organization of herself. Had Mai had a different boyfriend, even if she liked him as much as she liked Zuko, I think she might have let him go the same way she did Tom Tom. Zuko just brought with him a lot of things that challenged her apathy--and that was maybe the very thing that attracted her to him.