Historically, although it has diminished greatly and is increasingly confined to the areas you described (Quebec and neighbouring regions of Ontario and New Brunswick), French influence is relevant to almost all of Canada and even a large part of the United States as well.
There are established French Canadian communities in 9 of 10 Canadian provinces that have been where they are for 100 years or much more. In many if not most provinces of Canada that are today largely English-speaking, French speakers were actually the original European settlers. The one possible exception is British Columbia, and even there there is a historically French Canadian neighbourhood in the Vancouver suburb of Coquitlam that will be 100 years old in 2009.
The first permanent French settlement in the Americas was in what is today the province of Nova Scotia, from which the French Acadian population was expelled by the British starting in 1755. Many of these deportees eventually ended up in Louisiana and are the ancestors of today’s Cajuns. Other deported Acadians ended up in areas nearby that were either largely uninhabited (today parts of New Brunswick) or still under French control at the time (today’s Quebec). Nova Scotia today has a small French-speaking Acadian community of about 40,000 that makes up around 4% of the province’s population.
Outside of the "core area" you described (the province of Quebec, plus northern and eastern NB and eastern Ontario), most of the French Canadian communities outside Quebec struggle with assimilation to the English majority that surrounds them. In most provinces (including Ontario, which has a large and diverse population), they generally make up between 2 and 5% of the provincial population. In New Brunswick, which is relatively small in population and geography, French speakers are about one third of the population and actually dominate large areas of the province.
Opinions may differ on this (and you’ll likely hear other views on this), but the fact is that French cannot be said to be something that has been foisted upon the rest of Canada by Quebec. In the 100 years between roughly 1860 and 1960, much was done in order to eradicate the French language from the areas of Canada that are outside Quebec (the French Canadian population in Quebec was too large and organized for this type of scheme to work effectively there). Then, in the 1960s the separatist movement started getting active in Quebec, with the main argument being that "they may be trying to stamp us out in the other provinces, but the buck stops here at the Quebec border".
The Canadian federal government, and to a lesser degree, some of the majority English provinces (especially New Brunswick) tried to counter this separatist argument with greater support for their French-speaking minorities. This has been a mixed success, with pretty much only the communities you mentioned in Ontario and New Brunswick strong enough for the bolstering since the 60s to give them a chance of survival in the long-term.
Regarding the French influence in the U.S. this is obviously seen in Louisiana, but also in places like Detroit and St. Louis and many other smaller locations which were originally French settlements, and in place names given by French explorers like Coeur d’Alene and Boise, Idaho, Butte, Montana, Fond du Lac, Eau Claire, Des Moines and, of course, the classic favourite: the Grand Teton Mountains! (For fun, look up what that last one means!)