There are a lot of different systems - obviously nobody in their right mind wants the US system.
We're in Germany and we each pay 15.5% of our income - deducted from salary the same way our compulsory unemployment insurance, care insurance, pension insurance and income tax and solidarity tax (all separate payments) are.
Obviously this is a massively different € amount from person to person.
It's theoretically impossible not to be insured (and certainly illegal). Children are covered by their parents for free (in some circumstances if they're in full time education and not earning up to age 27). "Dependent" spouses (earning under 500€ per month or so) are covered for no extra charge on the "main earner's" state policy. Unemployed people and people in receipt of state pensions without a working spouse (including anyone medically retired) are covered by state payments.
The main problem is that people earning over a certain amount are allowed to opt out of the state insurance options (there are multiple state options) and be privately insured. The "I'm alright Jack" high earners do this because it's actually cheaper for them when young and healthy and they get some preferential treatment (there are doctors in cities who only take private insurance, though it's not really a phenomenon outside big urban areas, and some policies cover things like IVF more easily, and most give private hospital rooms and other luxuries accommodation wise).
Then they get old and less healthy and perhaps their earnings reduce or other outgoings are high and can't afford the suddenly higher premiums and want back into the state system which they opted out of because they didn't want to "pay for everyone else" with high premiums when they were young and healthy - and they aren't allowed back in.
On the less karmic side self employed people have to be privately insured, which is just stupid especially as there are plenty of low earning self employed people - cleaners for example (which is why you can't get an above board, doing everything legally and happy to provide bills/ receipts cleaner for less than 20€ per hour minimum...)
Also the private insurance is a real problem in some cases when children with serious chronic illness are on a parent's private insurance or have their own private policy and become adults as there are situations where it's near impossible to get them onto their own state insurance policy and their own private policy is unaffordable.
So it's a good system but IMO the private insurance shouldn't exist - everyone should have to pay their 15.5% (ish, there are some differences between state options but they're broadly similar) state insurance.