@brian.shapiro: The 16th amendment wasn't even legally ratified. You really don't need a constitutional amendment, you just need hundreds of million of sheep, in order to keep the grass level in the pasture.
Anyways, I prefer taxes/fees that can be closely associated with activities; for example, look at tollroads. I have no idea how this idea could be extended to health care. The best thing I can think of would be an array of sales taxes, taxes on food and risky behavior. But the last thing I want is to encourage the establishment of a nanny state or the collection and misdistribution of other people's money, even if it is in their own self interest.
If you take 312,000,000 people and take about $3,200 per person per year, you get about a trillion dollars. I'm not sure how high you would have you make a sales tax, but it could be figured out. At the end of the fiscal, if there is anything left over, send everyone a check proportionate to their annual receipts. If at any point, the money runs dry, start a ASPCAish pledge drive with pictures of pathetic looking faces on every newspaper and or website until we've got enough to cover sister sue's knee surgery and the associated costs of the campaign.