She says she's going to move out today.
Is it right for me to be concerned where she's going, because I seriously doubt she found a place by now, or should I just ignore it? Half of me doesn't really care, and the other half wants to make sure at least she isn't exactly living in her car.
I mean, you're concerned about her. And I'd hope that you would be.
As much as I agree with other people who say it's important for you to get some distance from this person -- and important for her to do the same with you -- I do want to caution against an attitude of "there are lots of fish in the sea" or whatever.
This person trusted you enough to confide in you that you are one of the few people in her life that she feels actually cares about her. Without knowing her I can't say if that was just her being manipulative or if that was a sincere, earnest statement. Either way, having that said to you places a tremendous responsibility on your shoulders. That's a big weight to carry, especially with somebody that you maybe haven't known that long. In some ways it's not fair of her to put that on you. If being there for her makes you unhappy, then it can't really work.
Still, I think it's probably important for you to try to communicate to this person that while you feel like you need to have some distance from her for now, you still care about her and you want to be there for her, eventually, when you're both ready. Otherwise, if she feels like she's really laid herself bare and trusted you as a person she can depend on, and you become cold and turn her out, it may ruin any chance there is that you can be friends in the future.