I was searching a little more on the subject and found an economic reason for public health care.

The problem here lies with a term called "market failure" where the market fails to function as it should as a free market.

In this situation, the problem of information assymetry from the customers side. When you receive treatment, the doctor knows the ailment, which drug will suit you best and how many more appointments you need. When the doctor earns money per appointment, he certainly has a reason to make more appointments than is needed, and if he is a "bad" doctor, give a drug that is not fully efficient so that it takes you a little longer to heal. That paradox is most famous from the psychology world - you go to a psychologist, usually, to solve a certain problem, and the psychologist faces a problem - the quicker he helps, the less appointments you have with him and the less money he receives (and the same rule applies to mechanics, which most of us been down that alley before). In a world where everyone is honest, it is not a problem, but we all know it's not the case here.

When the doctor knows he won't receive additional income from many appointments, it is likely he will give you the best treatment so that he has "less work".

In addition, Adam Smith reckoned that healthcare and education are two things that benefit the society more than it benefits the private person. For example, one might feel healthcare is worth x$ a month for him, but for the entire society it's worth far more: income tax, labour, not infecting other people (that in itself is worth more than what it is worth for him). So each and every person feels wants to pay a certain amount of money for healthcare, but since for the country it's worth much more, it creates more resources to the country to actually subsidize the difference between what it is worth to the society and what it is worth to the private person, thereby increasing the societal surplus.

The following link is to the article Social Choice and Individual Values by Nobel Prize winner Kenneth Joseph Arrow, where he explains in much detail the situation with its pros and cons. It is a rather long read, but a nice thing to look at if you ever have time.