I have little faith based upon a long history of individuals with little compassion taking advantage of the general populace. Just look at working conditions and employee rights in the 19th and early 20th centuries for what happens when the market is left to itself to "self-regulate" as it were. I shouldn't have to say it, but no I'm not saying the government should also just manage everything for us. But there are very real reasons much of our current legislation was enacted in the first place. The matter of the correct balance is of course, always up for debate.
Companies have a long history of offloading negative externalities onto the public and/or environment in order to increase profits. Hence why government stepped in to prevent them from just discharging toxic waste or sewage into waterways and into the air. Look at China's air quality issues atm for what happens when there are no controls.
As for healthcare itself: You're already paying for the people without healthcare. Billions of dollars are written off every year from ER visits by people without insurance or the money to pay for it. That money is then recovered by raising YOUR insurance rates. IMHO I would rather have everyone paying their fair share, than footing the bill for someone else.