The only difference is that you have a choice in whether you want to pay the money in exchange for the coverage. Which is exactly the point.
The poor don't get out of paying it by having nominally low personal tax rates, because all the other taxes still apply to everyone. When you tax a business and it raises prices or can't give employees higher wages, that comes out of the pockets of the poor the same as everyone else, which is what makes them so poor to begin with. Remove $500 in spending of which $250 were real services, then pay $250 for the real services yourself, and the result is a $250 surplus -- which can go to the poor as much as anyone. They are, after all, the largest beneficiaries of lower unemployment (resulting in more jobs with better wages) and lower prices (because they spend more of their income).
The poor don't get out of paying it by having nominally low personal tax rates, because all the other taxes still apply to everyone. When you tax a business and it raises prices or can't give employees higher wages, that comes out of the pockets of the poor the same as everyone else, which is what makes them so poor to begin with. Remove $500 in spending of which $250 were real services, then pay $250 for the real services yourself, and the result is a $250 surplus -- which can go to the poor as much as anyone. They are, after all, the largest beneficiaries of lower unemployment (resulting in more jobs with better wages) and lower prices (because they spend more of their income).