It really is the "everyone" that annoys people. I've asked conservatives to explain to me how insurance works as a business model. They pretty quickly realize two things:
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Monthly premiums of everyone cover the needs of anyone in need today. (Some seem to think it is like a bank account, but pretty quickly realize why that doesn't make sense. Framing it around car insurance instead of healthcare tends to help if they are struggling here.)
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As a for profit business, they need to collect more in monthly premiums from each customer than they will provide that customer with benefits. Otherwise, the business is not profitable.
Once they understand these two facts, then they are ready to understand why government sponsored healthcare is optimal. It maximizes the number of people paying into the monthly premiums, while minimizing the profit margin because the government doesn't need to be run at a profit. Many disagree with that last point, but will usually be satisfied by phrasing it as "bringing in a minimal profit. Otherwise, we wouldn't want to be overtaxed, right?"