If you renounce your citizenship in that country, do you....?

No. You have to specifically apply to be a citizen of anywhere else, having fulfilled all the conditions to become a naturalised citizen. In Canada that includes having lived there for 3 years. Each country has its own citizenship laws and if you want to be one of their citizens, you have to comply with them.

If you renounce citizenship without having another one first, you become stateless and countries won't usually let you do that. I know the US won't accept your renunciation if it makes you a citizen of nowhere.

Edit - brucec83 isn't correct there: you don't necessarily always have the right to live in the country where you were born. That is true of the USA, because anyone born on US soil is a US citizen, but the USA is unusual. The primary method of becoming a citizen in most countries at birth is to be born there AND have at least one parent who is a citizen there, or simply to have a parent who is a citizen of that country wherever you are born.

Thus if two illegal immigrants had a baby within the borders of the USA, the baby would be a US citizen just because of where it was born, and that makes it harder to deport its illegal parents. This could not happen in the UK as British law additionally requires that it have a British or legally permanent resident parent to be a British citizen - that's why we Brits changed the law in 1983 and why most other countries work that way - and the baby would not have the right to live in the UK as only British citizens have that right.

Germany is an even more extreme example. Being born in Germany has no connection at all to citizenship: it goes entirely on whether you have a parent who is a German citizen.

You MUST first apply for and be sworn in as a citizen in the new country first before you can renounce your old citizenship. The old country will NOT allow you to renounce if you do not already have another citizenship in place. Countries do NOT want to create people who are stateless.

And NO the scenario you describe - there was a draft in the US, and people began fleeing to Canada to renounce citizenship - does NOT give the draft dodger any automatic canadian citizenship. The draft dodger has to apply for PR status, be approved and then live in Canada for 3 years before they become eligible to apply for canadian citizenship.

surely you dont expect any refugee who asks for asylum in the USA to be given automatic US citizenship do you? Of course not. Only one country that I know of gives out automatic citizenship in one specific situation. That country is Iran and thats only when you marry an Iranian citizen.

The new country has to accept you first otherwise you are stateless.

Try to think this out logically now: If all the illegal immigrants from Mexico renounced their Mexican citizenship because they would automatically become USA citizens, our illegal immigration problem would be solved. It doesn't work that way!

No. Unless you hold dual nationality, you become stateless (and as a refugee, qualify for a UN Laissez Passez) unless the receiving country allows you to naturalize. Since you always have the right to live in the country in which you were born (even without citizenship), the US generally won't allow you to renounce citizenship unless you are outside the country and have another available to you.

I sometimes want to bang my head against the wall