To receive health care in Ontario, you visit a doctor. Then you are triaged based on your needs. If you have a non-life threatening issue that requires an in demand specialist of some kind, you are likely going to have to wait a while (several weeks, or even a month or two) to get treated . . . although you will eventually get treated and will get good treatment.
Rich people don't want to wait for treatment generally, so to prevent them from simply buying private health care and effectively forming two tiers of health care, it's not legal for private hospitals to run in Ontario. Except when that bill was introduced several privately funded hospitals already existed . . . they were grandfathered in and allowed to continue running. This Shouldice clinic (incidentally, it's supposed to be one of the best hernia repair place in the world) was one of them.
It exists in kinda a weird place in our system. Much of the funding for the clinic today is public, but they do take private patients. Canadians can receive treatment there paid for by our public health insurance when recommended by their doctor, or they can personally pay the cost of surgery and get treatment there. The clinic has some odd policies (like refusing you treatment if you're over 20% of your ideal body weight).
So, it's likely that Jack Layton didn't know the place was private when he went there, and it's also likely that Rand Paul paid for his treatment at the clinic because he couldn't find a better US clinic.