Should it be a requirement? No. Should it be an affordable option for everyone? Yes. Some people do not believe in modern medicine, and although I do not agree with them, I think they are entitled to their beliefs and should not be forced to spend money on something they would not need, for example.
As it is, insurance is too expensive to be a requirement. If you can barely afford to keep a roof over your head and food in your mouth, than insurance would seem like a luxury, yes? No good having insurance if it means sacrificing essentials that will endanger your health.
We would not need required insurance if quality health care were affordable. How to offer affordable health care is another issue entirely. That would require more affordable technology and testing and more affordable doctors rates (which would require more affordable medical education, because you cannot get all of those student loans and then not get a job that will pay enough to make the education worth it.
But I digress. The bottom line for me is that insurance could never be a requirement, but should be option.