Everything depends on the state you are living in. For an example if you live in DC, they can get health insurance as low income residents, but not in VA. However, the catch is that when you filed the AOS; you have stated that you will provide for them and they won't be on state welfare. Legally, the state can ask you to compensate for any benefits received. But I have never seen the States running after someone for benefits paid. You have to let them file the taxes on their own, so they qualify for the benefits though.