When I filed my taxes in March, I asked to appeal my health insurance penalties (state/fed). I paid the state, as I was asked to, but have not received any paperwork yet.

there is no federal requirement to have health insurance. Only some states require it. so you would have to check with your state. 

if you were covered by the Affordable Care Act (form 1095-A/8962), you would have to pay back (Federal) excess advance premium tax credits, subject to limitations, This is not appealable.