The Americans tried but the Canadians were unwilling partners. Canada had primarily been settled by the French, who were of course Catholic. As a result, the Catholic clergy held great power, and most of the power holders were Catholic. In 1774, before the Revolutionary War started, and after the Boston Tea Party, the British passed the Quebec Act. It was designed to assuage some provisions that may have led Canada to join the US rebellion. Among other things, it granted a lot of territory (including a lot of the US Midwest) to Quebec, removed a requirement that allegiance to the protestant faith be required for eligibility to hold office, allowed French law in a number of areas, and restored the right of the Catholic Church to impose tithes. The result of this was that most of the power holders in Canada sided with the British, at least officially. The Americans led propaganda campaigns into Canada and invaded Montreal, after which they tried to invade Quebec, but failed. Without people joining the cause, which the Americans expected, the effort fell apart.
Nova Scotia almost joined the revolution. A 1942 historical novel by Thomas Raddall ("His Majesty's Yankees") does a fine job of illustrating just how much sympathy there was for the revolution in Nova Scotia. Most of the people living in Nova Scotia were from New England after all.
Ironically, the progression of hostilities leading up to the revolution helped push Nova Scotia into the loyalist camp. People loyal to the crown kept moving to Nova Scotia seeking safety as the conflict ramped up, tipping the popular balance towards Britain.
In the 1937 book "The Neutral Yankees of Nova Scotia", John B. Brebner argues that Nova Scotia's geography, being entirely surrounded by sea, made it easy for the British to secure, thereby making it an appealing place for any loyalists fleeing the revolution.
As far as the French colony of Quebec goes, they used the revolution as an opportunity to extract significant guarantees of autonomy from the British by threatening to join in the revolution, which was enshrined in the 1774 Quebec Act. No such guarantees for entrenching language, religion, and medieval social hierarchies (still practiced in Quebec) were on offer from the rebelling colonies.
Thus codified the special legal exceptionalism of Quebec has persisted to this very day. Every time Quebec threatens another referendum on secession Canadians can have fun blaming the Americans for causing the perpetual rift in their society. The British never would have granted special autonomy to Quebec if they weren't desperate to keep even more colonies from rebelling.
I'd like to build on /u/secondsniglet's comments on Nova Scotia, having just written my honours thesis on Nova Scotian neutrality and the printing press and am going to be starting my MA in history next year to continue this research. Raddall and Brebner are both sources that I used but both are also quite old sources. There has been a lot of work done on the question of Nova Scotian neutrality since the 1940's and I would like to through a bit of it here. But first I need to give a bit of background.
Firstly I should establish that at this point Nova Scotia encompassed New Brunswick as well. New Brunswick and Cape Breton were created after the war to divide NS in case of revolutionary unrest. PEI was just being settled and was so sparsely populated that Revolution was not really in the cards for the island, although it was raided by American privateers during the Revolution, just as NS was.
As such the question "Why didn't the Canadian colonies join the American Revolution?" can be safely divided into two: that of why Quebec didn't join, and that of why Nova Scotia didn't join. I would be interested in any account of Newfoundland during this period but have not encountered anything more than the odd footnote on the topic of Newfoundland and the Revolution. I'll leave the question of Quebec to one better versed in its history and stick to trying to answer the Nova Scotia question.
I peg the beginning of the American Revolutionary Crisis as being 1765, with the passage of the Stamp Act that did so much to enrage and alienate North American newspaper printers and more importantly their subscribers. Sloan and Williams write in The American Revolution and the Press that this unified mass media opposition to the Stamp Act was unprecedented in history, and the perception of unified outrage that it created was a huge factor behind getting different British North American colonies to unify in opposition to Parliament's taxes.
In the 1760's the French had been shattered and expelled from Eastern Canada. To fill the vacant French Acadian lands Governor Lawrence issued a proclamation promising free land to any New Englanders that moved to Nova Scotia. Furthermore in 1758 he promised to form a Legislative Assembly, with two members elected from every township of fifty families or more, in order to attract the New Englanders who were used to their own democratic, decentralized form of township government.
Before this NS had been ruled by a central, appointed Executive Council that served at the pleasure of the royally appointed governor. Giving townships representation and some modicum of authority was an about-face in Imperial policy, which had previously been against developing Nova Scotia into a "New New England", as the Imperial government found the decentralized New England system irritating, as local township governments possessed the authority to impede the efforts of London authorities to impose economic and political agendas on the colonies.
My favourite book on NS during the Revolution is Fault Lines of Empire by Elizabeth Mancke, and she singles out township government as the point of divergence between New England and NS that sees the first revolt and the other remain neutral. She convincingly argues that Lawrence's system only appears to be similar to the decentralized New England system, but that the Executive Council actually still maintained sufficient control to stymie local government in NS, in large part due to the Executive's ability to use discretion in granting townships, and that it is this more than anything that keeps NS neutral. It is a great piece of historical work, very focused and extremely methodically researched.
But the plan worked and New Englanders flocked to NS in search of homesteads. The upshot of this is that by the time the American Crisis began over half of the population of NS was "American" in origin, that is they had immigrated from New England under the terms of Lawrence's Proclamation, bringing their culture, political views, and contacts/family connections to New England with them. This raises the question: With New England being a hotbed for revolutionary sentiment, and with New Englanders having a strong tradition of political participation, organization, and petitioning through their strong township system, why didn't Nova Scotia organize and revolt as the other colonies did? I will give my answer to this question later as there is some work I need to do first but I hope that this post intrigues a few people to a mostly overlooked story of the American Revolution.
Although much later, the Economic Union Party was formed in the Dominion of Newfoundland in 1948 and proposed free trade and customs union with the US. The EUP was attacked as anti-British and disloyal to the monarch. Newfoundland joined Canada a year later.
Better question is why didn't the British Caribbean colonies.