If a person has little or no out of pocket costs for doctor visits and medical procedures is that person more likely to "run to the doctor" for every little problem?
Although we cannot predict the behavior of any individual consumer with any accuracy, we can say that, in general, a person with lower out of pocket expenses for each visit to the doctor will consume more of those visits than someone with a high out of pocket cost. This is a basic idea encapsulated in the law of demand.
The law of demand tells us that, all other things being equal, a decrease in the cost of a good or service will lead to a higher quantity demanded for that product. This makes sense intuitively because the average person will consume more of a good if it is cheap than if it is expensive. There is no reason to think that medical services will be exempt from this law.
Of course, there will be some people who will not go to the doctor for every little thing. They might feel that it is their civic duty to refrain from consuming too much in the way of medical services. They might feel that they are tough enough to not need medical attention for minor things. But, on average, a person who does not have to pay much for medical attention will seek more of it than someone who has to pay a lot.
This is one of the problems with our health care system. We typically have low out of pocket expenses for various medical services and therefore we have little incentive to forego trips to the doctor.