Some things are compulsory in society by necessity. Laws and taxes are two of them.
Whether or not personal health care that also benefits society at large, including employers, should be compulsory depends on your point of view. I can only speak from my own experience in the UK. The NHS was created just after I was born and I saw all of its growth pains. It works extremely well but, like any other large structure, Isn't perfect. First line care is from a local doctor. They usually work in group practices and have their own premises and nursing staff for dressings and things like that. The also have pre- and ante-natal care. The doctors are self-employed, contracted to the NHS. For emergencies and operations, you go to a hospital. Some have specialist wings for heart operations and cancer treatment, for example. If you have an urgent problem, an NHS ambulance with trained paramedics will come to you. They will give first aid and cart you off to hospital. Each local health authority has professional management and budgets. Now, because of long waiting lists in the past, the government has set service standards, enabling patients to know what they can expect and where to complain if things aren't right.
I think that the service is amazing and wouldn't have been without it. Despite the tax cost to me and my employer, I always had enough cash left for food, home, car, vacations, beer, savings and all the usual stuff. I also had private health care from time to time when employers offered to pay for it and used it once for my son. It's much more expensive than the taxes I paid for the NHS and State pension and the treatment is no better.