OP, many schools will provide a health insurance plan (possibly with option for family to go on it as well). You should ask about this when you apply/interview. If your assets are low enough, the children at the very least should qualify for Medicaid.
On a larger philosophical level, if you are currently living at a middle class level, but your resources will be spread thin to the point that you envision requiring welfare assistance, I do think it's fair to question whether putting your family through the hardships and privation of you attending medical school is "worth it." Not to say that you can't or shouldn't go to medical school as a parent, and obviously everyone who does go to medical school has to make significant financial and personal sacrifices. But there is a huge difference between doing it with adequate resources in place (living in a city where you have family to provide childcare, for example, or waiting to attend medical school until you have enough resources to help support your family at a decent standard of living, plus/minus any needed loans for the four years), and doing it hoping that government programs will be there to make up the shortfall so that your family isn't living below the poverty line. Of course, if you qualify for an assistance program, and you need it, then you do what you need to do. I don't think anyone could begrudge your children in particular having adequate health care while you're in school. But considering how the rules of government programs are constantly changing, and especially with health insurance (which is your top concern) being up in the air at the present time, it may be worth considering whether right now is the best time for you and your family to have you quit your job and go to medical school. If medicine is your dream, I definitely don't envy the position you find yourself in. :-/