My daughters boyfriend was in a situation where his mother, sister and himself had to leave their home because dad was an abusive drunk. We took these three in and gave them a place to stay for a few days. On the third day the mother abandoned her son on us and refused to tell him where she was. He stayed with us for ten months and neither parent ever gave him any financial support. During these ten months he turned 18. My question is, is this child abandonment and can I go after child support for the months that he was still considered a minor child.

No you can't go after support. And while it is abandonment it is not criminal abandonment. Criminal abandonment would be when a child is left without any adult supervision. That clearly didn't happen. If you were the child's legal guardian or custodian, THEN you may have a case for support. But under the circumstances you list, no.

You could have made him contribute to the household. When the mother left, and refused contact, you could have called family services and have him placed in a shelter. That you offered him room and board was a kind and generous act, but not one you could be financially reimbursed for. Unless the boy decides to do the right thing and contribute.

You could have when it happened, called CPS and they would have taken child in. You could have filed for temp guardianship in the court ( and you did not do that) Had you filed for guardianship then you could have tried to sue for support.

You could try, to sue them in small claims court for the cost of meals and other money you spent on the child ( did you keep receipts ?)