Canada and its citizens had no involvement in the burning of the White House nor the battle or bombardment that took place shortly thereafter in Baltimore during the War of 1812. While you're correct that Canada wasn't its own established country by this point, it was its own territory similar to how New York or Maryland weren't a part of a separate country in the 1750s, but it would not be fair to say that Maryland or New York were fighting land battles in Europe during the Seven Years War. Regions had their own identity even if they were British subjects.

The War of 1812 is a war that seems to get lost amongst American educators in this country, and it's a bit peculiar why. The War itself is often viewed as a branch of the Napoleonic Wars of France and Great Britain, although Americans played a very distinct role in this part of the broader conflict. The war's origins date back to struggles around trade blockades that the British imposed upon France, which the United States deemed was illegal in the eyes of the international community. Minor conflicts ensued, with Britian's Navy capturing and impressing American sailors into service and the US ships eventually retaliating, as in the case of The Little Belt incident off the North Carolina coast in which a US ship killed 11 British sailors and wounded 21 others. Many escalating incidents continued until June of 1812, when President James Madison signed a declaration of war against Great Britain.

America, from the start, almost seemed poised to annex new territory from Canada. In the first major battle of the war, Americans attempted to take control of a region in present day Ontario, in what is known today as the Battle of Queenston Heights. The battle poised US soldiers and militias against British regulars, Canadian militia, and Mohawk Native Americans and was a sound defeat for them.[1] What's interesting to note is that even in Canada, British Army was distinctly made up of British soldiers who grew up on the British Isles. This created a problem for the British, since young British Privates soon started seeing better opportunities in both Canada, but especially America during their deployments in the war. In his pulitzer-prize winning book, The Civil War of 1812, Alan Taylor points out that British regulars started deserting in high numbers even before the war began.

The empire also lost men along its Canadian border with the alluring republic. A Canadian military historian concludes, "Nowhere else in the empire did so many men flee the colors so easily." Deserption depleted a small force already spread thin...

Both British push and American pull promoted desertion. deserters fled from the tedium, low pay, bad food, and harsh discipline of the British army, while they ran to the abundant meat, cheap liquor, and beer wages of the United States, where a common laborer earned four times the pay of a British private.[pp. 107]

Taylor goes on to explain that the majority of British soldiers serving in Canada at the time were Irishmen from very impoverished backgrounds, saying "Impoveresihed Ireland had become the prime recruiting ground for the British army."[pp. 107] While Canadian soldiers were not the ones who made up the British army, they did have their own militias that not only were fielded in Canada, but preformed well in field, including the Battle at Queenston and Fort Erie.[ pp 199].

Similar to the Battles up north, the British military's movements in Washington D.C. and Maryland likewise was fielded by British regulars made up of soldiers who were recruited from the British Isles. Because Canadians did not make up members of the British Army, nor did a Canadian militia arrive at Washington during this period, it would not be possible to say that Canada or its citizens had a hand in burning the White House, U.S. Capital, or partook in battle at Fort McHenry a few weeks later.

Alan Taylor. The Civil War of 1812: American Citizens, British Subjects, Irish Rebels, & Indian Allies. Vintage Books. pp 189.