Please contact a lawyer who deals with child custody and child support to assist you enforce your legal rights in this matter.
The rules are no different for an illegal immigrant, child support and child custody are fully enforceable.
The challenge you face in terms of child custody is ensuring he is not a flight risk.
Keep in touch with his family, this is always a backup strategy.
You may still file a petition for support and custody, however, it may be difficult to enforce the court's order if he at some point is subject to deportation proceedings.