Within the last year I have had one difficult child spend one week at psychiatric hospital (aka Behavioral Health Center) due to depression & suicide ideation and another difficult child spend two weeks at a shelter for troubled/runaway teens and 4 days in Juvenile Detention. Our circumstances might be different than yours, but BOTH of my daughters did insist at one time or another that they did not want to come home. Two thoughts to add to this discussion:
1) In one case I am absolutely certain that the difficult child was angry with me and fearful that I was rejecting/abandoning or 'giving up' on her. It was a defense mechanism for her to say that she did not want to come home, I am not her mother, she didn't want to be in this family, and wants me out of her life. In order to feel power and control, she wanted to reject me first and emphatically, before I kicked her out for good. In the heat of the moment, those words did hurt my feelings, but when I look at the big picture, I understand that this is only her anger and FEAR talking, and that she really does not mean it. However, my children are both adopted (formerly foster) children, and so they may have some paranoia and insecurity about the permance of our parent-child relationship.
2) We know that these (and most) children often blame us for all of their problems. This is due to lack of maturity, experience and responsibility. While you and I know that everything is NOT 'all our fault,' these immature and illogical children seem to think so. It is *their* behavior that we would like to change, but this does not happen easily Autism Spectrum Disorders (ASD) we get 'stuck' in the same patterns by which wer try to correct them, which are not having the desired effect. In fact, since we are the ones who acknowledge that there is a problem, maybe we should STOP and consider whether some of our parenting style is contributing to the problems. Without actually accepting blame for our difficult child's bad behaviors, let us stop and consider if there might be something in our parenting style that can be improved, so that the difficult child would want to come home. In my case, I have had to realize that I was suffering from depression (and sleep apnea), and my terrible mood and exhaustion were affecting my ability to parent. I had developed a negative communication style (scolding, lecturing, nagging, ven sometimes name-calling or insults). Family counseling (all three of us together) has helped us significantly, as well as behavior charts with rewards. And when the difficult child's see that I am working on improving my own parenting style, they seem more willing to work on their behaviors too.