Because the group of men fit to fight such a war would rather rebel against the government than fight a brother war. From lowest recruit to highest general.
Your use of "brother" is apt. There's a Ukrainian joke that goes something like:
"A Ukrainian man and a Russian man are walking together. They happen upon a $20 bill on the sidewalk. The Russian man says, 'Let us share it as brothers'. The Ukranian man says 'No, let us share it equally'".
The only realistic scenario I can think of when your American "brothers" would go to war across the border is if the Canadian government commences war against its own population. Then I could see the US government intervening, or US fighters independent from the government taking sides in Canada.
The problem is that most people think of these scenarios as something that happens overnight, when in reality consent is manufactured over time. There's a reason you don't microwave a frog.
A lot of Canadians talk big talk about some sort of insurgency like Iraq, Afghanistan or Vietnam but all those places have borders with other countries that can enable smuggling of supplies to the resistance.
Canada will be blockaded and after a period of cold and hunger the Canadian people will give up.
Then simply marched back home and said “stop being so stupid.”
About 85% of all Canadian militiamen remained at home when called up in 1812. In 1812 and 1813, British regulars and Indigenous warriors (from both the U.S. and Canada) invaded Michigan and Ohio, but didn't get any further than that before the U.S. counter-attacked.
What makes you say that?
I could see heavy protests, even violent protests, as it's not something Americans want.
I'm not sure I could envision any semblance of an actual civil war, though, but perhaps I'm underestimating things.