I think private healthcare providers would still be able to succeed under universal healthcare. It could be a little more challenging but I think it's possible. Even with universal healthcare there will be individuals who would rather go to a private provider than the governmental one, and will do so if they have the money available. These people would support the private healthcare providers. Additionally, private providers would have competition between themselves because they will be trying to get the wealthier individuals who use private providers to come to them versus someone else.