I don't think that single payer universal health care is important to preventive care because premiums and the quality of health care shouldn't have a correlation. I believe preventive care can increase if research and medical school subsidies are increased for student who are able to qualify for medical school.
No, free universal health care is not important to preventive care, because free universal care does not provide the best service. As more people want services, there is only so much money to pay for them. So what used to be a screening every two years under private insurance will now become a screening every four years. We will all have less care.