Supplemental private insurance isn't required in Canada (other than if you are travelling out of the country, of course). I have never heard of anyone requiring private health insurance, nor have I ever seen it advertised nor do I believe it is even offered as an option here.

Unless you are talking about dental care or optromitrist (sp?), they are not covered by public insurance (eye doctor visits used to be, but were delisted a few years ago). Those things are covered in the benefits packages for most full time jobs though, and there may be private insurance available for them as well but I'm not sure.

The costs of visiting specialists etc is covered by public insurance. Drug prescriptions are not fully covered but are highly subsidized (most drugs only require a flat "dispensing fee" to recieve),

Canada does have issues with illegals (or even more problematic: refugee claimants, often fraudulent) although I agree it isn't nearly as severe as it is the the United States.

I have not read the Fraser institute report (but I will), although I will caveat before I read it that the Fraser institute is recognized as being extremely libertarian bias, beyond the point of objectivity.

I agree total healthcare costs would increase, but that is a consequence of expansion in the number of people covered.