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Thanks for your question. there are two situations here where you may owe for support. If the court gave custody to the state and set support for you to file you may owe arrearages.
If they did not do this at the time but federal dollars in the form of medicaid or other benefits were paid on behalf of the children then yes they may attempt to set some support repayment amount that would allow for recovery of these expenditures. One of the functions of Idaho Child Support is to do just that so that the funds if recollected can be reused for more children in the future.
Attached is the web page for Idaho Child Support/Social Services. Note the information about medicaid, this is probably where the expenditures were made for coverage for the child/ren and they are trying to recover some of the expenditures. I understand how this came as a surprise but it is within the law for them to do so. Hopefully you can negotiate something you can live with here..