There is no "Euro" healthcare system. There are many different solutions. They range from completely public systems to highly regulated private industry. Universal healthcare does not necessarily equate to a public, one payer system. There is a good argument for treating healthcare like a utility. Many utilities in the U.S. are privately owned but operate under strict government regulation and oversight. Profits are often restricted to reasonable percentages.
The goal of insurance is to distribute a small number of large costs into a large pool of small costs. Currently the insurance industry is allowed to have their cake and eat it too. Not only do they get to charge an "average" rate for the average person, but they get to identify the non-average persons and jack their rates. There are many simple laws that could be enacted to make the market more free. Off the top of my head:
1) Make switching from one insurance company to another easier by making coverage instantaneous with no lapse.
2) Eliminate denial of coverage for pre-existing conditions. Previous medical history cannot be taken into account when acquiring insurance customers. No customer can be turned away.
3) Enforce a one price policy where an insurance company is required to grant all insurance at the same price for all customers with very few modifiers. There should not be a situation where a large corporation negotiates a lower rate than what is paid by a smaller company or an individual.
4) Modify 3) above to allow an upcharge for certain types of conditions that are under control of the customer and result in better health, namely things like smoking and body fat percentage, even exercising. The upcharge needs to be a borne by the customer directly, not hidden by the payments made by an employer. Example: An employee is covered by his employer but he has a greater than 20% body fat percentage so he has to pay out of pocket $100 per month.
5) Related to 4), force more costs on to the customer. If an employer currently pays $10K per year for an employee's healthcare, it is better that the employer pay $6K to an insurance company, an extra $4K to the employee, and the employee pays the $4K to the insurance company as services are rendered. Those who see the doctor for every sniffle (or H1N1 virus

) need to have an incentive to suck it up and be a man.
6) Get rid of punative damages in malpractice suits unless something was done deliberately or with malice.
7) Cover all members of Congress with a plan than mimics the medical coverage of the average American.

More devious would be to determine average plan for each quartile of the population and let each member of the legislature draw by lot which of the four plans he will be covered by for the year.