Why would she ask you to co-sign....You have answered this question already. Neither she nor the new boyfriend have the credit nor income to finance a car loan; you do. Assume that the person asking was s sister, niece, cousin, one of your parents, or someone you hardly know. Would you co-sign given an equivalent record of bill paying, living within one's means, sense of general responsibility, sense of general obligation to you/for paying off a loan, general stability, income level, and resource allocation decision making within the union? Chances are no, no, no, no. Chances are, if you felt the need, obligation, or want you would forego co-signing and simply buy a vehicle as a gift or make it a direct, person to person, loan. This is all the more critical since there are ways that many have turned the simple co-signing of a loan into to the acquisition of additional credit, loans, and credit card based upon the credit of the person who generously co-signed for a simple loan.
Among other things, it sounds like the ex doesn't know how to assess a relationship of any kind. This includes one with a guy/mate, money, spending above one's means, and probably a lot more than this. It does not surprize me that she wants to move back in with you for stability, emotional, care taking, and financial reasons. The real question is: Is she able to become a contributing partner of the type that you would be willing to spend the rest of your life with? Do you even have any shared goals in life? Can she contribute to the relationship whether she ends up ever working or not? Suppose you get together and she never ends up working in the workforce, is she even capable of recognizing how she can support your career towards the support of the family or is she more likely to continue to be a taker in the relationship?
You have some hard, end of relationship assessments still to be made. Just what is she capable of, what are your obligations towards her, are you able to allow her to flounder homeless, and just what are your limits with/for her should you help her in any way?