As has been correctly noted, the US and Canada are the only two countries in the world today that grant birthright citizenship (i.e. citizenship just by being born in the country). I have looked into this matter out of curiosity and found out that no other country grants birthright citizenship, except in very special circumstances.
Most western countries have adopted some UN charter on what's called "right to a nationality", which means that every person on the globe should have at least one nationality (i.e. citizenship) and that countries should work to minimize the number of people without citizenship, or "stateless persons", as they're called. Under this convention, in any western country other than US or Canada, the only way to get citizenship just by birth on the country's soil is if you don't inherit any of your parents' citizenships at birth. That is, if you would end up being "stateless" at birth, then and only then you get citizenship in the country in which you were born. Otherwise, no.
As for eliminating birthright citizenship in the US and Canada, there are difficulties. It's much harder to remove birthright citizenship in the US than in Canada because in the US it's provided by the Constitution so this would require an amendment to the Constitution. It's much harder to amend the US Constitution than to change a simple US law. Just to give you an idea how difficult it is, an amendment to the US Constitution requires passage in BOTH chambers of Congress (the lower House of Representatives and the Senate) by two-thirds majority vote. IF it survives this, then it goes to all the individual state legislatures for approval, and it has to be approved by three-quarters of the states (i.e. 38 states or more). Needless to say, this is a tall order.
In Canada, it's a different story. All they would need to remove birthright citizenship is to pass another citizenship law, like the one they just passed. That is not the problem though. The real problem is implementation of the new code of citizenship, once it's passed. Right now, a Canadian birth certificate issued by any province is adequate proof of citizenship but with birthright citizenship gone, the provinces would have to modify their birth certificates to indicate whether the newborn is a citizen at birth (by verifying the status of the parents). The problem is that the federal Government of Canada cannot bring all the provinces together on board to implement this change. I read an internal paper released last year by the federal government, which considered the possibility of removing birthright citizenship and it talked about this as being the problem: you cannot rely on the provinces to willingly change their birth certificate format to accomodate the new code. In fact, I just read last week that two of the provinces, Ontario and British Columbia, after having got a hold of the exact document I just referred to sent their strong objections to Ottawa over the proposed removal of birthright citizenship. And if you can't get the provinces to support you in this, it becomes very costly.
In terms of what happens to a family where the child is a Canadian citizen at birth and the parents are without status, the answer is Nothing that wouldn't have happened before the child was born. Parents will not get any legal status of any form and if they are illegal they are still deportable, same as before. This being said, there is nothing in any Canadian law that says a child citizen has to stay in Canada. If illegal parents decide to pack up and go home, they're free to take the child with them. Contrary to common mythology, the citizen child is not "property of the Government of Canada." But the child is a Canadian citizen, in the sense that if he wanted to take advantage of all the benefits of citizenship he would be able to (including the right to stay in Canada, to be schooled in Canada, the right to a Canadian passport, the right to sponsor his parents for permanent residence when he turns 18, etc. etc. etc.).
Also if parents decide to stay in Canada illegally to care for the child, they themselves are not eligible for any financial assistance from the government. However, the citizen child will be eligible for aid programs and since the parents are the legal guardians of the child, the government's money for the child goes into their hands. I used to see this all the time when I lived in the US. Illegal parents don't get money directly but indirectly they do because the government cannot deny benefits to a child when the child is a citizen, even when the child's money would go to the hands of parents who are illegal.
