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You could get a work visa, but there are only a limited number of these. Your chances of getting a work visa increases greatly if you have some sort of skill that could be applied to a job that is in need where you'll be living.
I do not know what kind of skills would be useful in Quebec, but generally in all of Canada knowing a skilled trade is very useful (electrical, plumbing, roofing, masonry, carpentry, etc..), having some sort of higher education (graduating from technical school or college for example), it's basically the same as getting one in the USA.
You could try and get a full citizenship but I'm not sure how that works. There are tests, and it takes time so I'm not sure if that option is for you.
You could get a student visa if you can afford to enroll in some sort of post-secondary course.

I would like to add though, that while we do have social health care it's not entirely free. We pay for it in taxes (as I know you know, I read your posts you're fairly smart), just be prepared for that when you get paid. I'm not sure about Quebec again, but minimum wage here is around $8 an hour. I make about $22 an hour, so I get around $1760, but only $1340 after taxes.

I will also add though, that the harder you're willing to work the higher your chances are. Our government has a way of...showing our appreciation of those who are willing to work in the oil field or mines by helping you get a visa or citizenship. It also pays incredibly well and requires next to no education, just common sense and physical work.
But again, Quebec, I'm not sure about. I'm actually pretty sure Eastern Canada would be one of the hardest places to get into as a foreigner. If you can, I would highly recommend thinking about Western Canada, if you go the oil field route you would be able to easily afford your own apartment and vehicle.

Oh and, not speaking French might hurt your chances quite a bit in Quebec. The first language in that province isn't English and in most occupations there priority goes to those who speak both languages or at least French.