You do not have to be a Canadian citizen to become eligible for health insurance. You only have to be a permanent resident. And national health insurance is a bit of a misnomer since each province controls its own health insurance. For instance, here in Québec it is the "Régie de l'assurance maladie du Québec" that is the government organization in charge of health insurance... www.ramq.gouv.qc.ca/index.shtml

To sign up at the Régie for an insurance card (called a "carte soleil") you must have been a permanent resident for at least 3 months-- Actually you sign up when you arrive, but they do not send you the card for three months unless you are a citizen of a country that has an agreement or an accord with Québec (such as France, etc.-- the US does not). I think it depends on the province how fast you are entitled to a valid health insurance card.

As for becoming a permanent resident can be anywhere from 1 to 3 years, I think.