Drivers Licenses aren't some privilege, they are supposed to be a safety check verifying that a person can operate a vehicle without destroying lives and property willy nilly. By making them a citizenship check you effectively ensure that no illegal resident's driving skills and road safety knowledge will be tested prior to them taking to the road.
Health insurance for all is the same kind of situation as the drivers license except instead of being about safety on the roads it is about financial responsibility and making sure that hospitals treat patients first instead of checking bank and credit card balances. Hospital ER's are required to treat patients first and figure out the bill later. That means that for many people in the US their primary physician is whoever happens to be on duty at the local ER. When they can't pay for the bill, which will be exorbitantly high because they didn't get treatment until it was an emergency and provided through the most expensive channel possible, the rest of us who do have the money/insurance to pay our bills actually take on part of that cost. I don't like the way Obama care was implemented, as it is a half-assed measure. He should have pushed through single payer federal health insurance and let the private vulture insurance companies meet the market for extra insurance.
I'll grant you that giving the right to vote to non-citizens is on the face of it pretty silly. Although if you look at it and realize that most citizens don't care enough to vote at all it's hard to see how it is worth refusing the right to interested people who live in the area, even if they do so illegally. And even in the event that you vote your lizard in s/he is still a lizard and likely to pay far more mind to any potential corporate sponsor than their electorate.