Personally, I think our right to health is more precious than our right to property or wealth. I'm not opposed to wealth by any means, and entrepreneurialism is a wonderfully fun enterprise, but they are lesser concerns than whether or not people are dying for preventable reasons. Ideally, a government encourages both health and entrepreneurialism by finding ways to pay people searching for more effective healthcare, then by rewarding the people who find it, but that's not what we're debating here.
The answer to "Should the government take my money to pay for somebody else's flu shot, if they cannot afford it?" is a near-unequivocal yes, for me, with the one condition being that I have enough money to afford that flu shot for them. Some people can pay for many more flu shots than I can, and I do believe that it's moral to request that they do so.
What I believe is that people innately have rights, regardless of what government regime happens to preside over them. The best governments are ones that are set up to protect those innate rights. To protect rights by taking rights away is a concept that doesn't make sense to me.