A Canadian taking a firearm into the US requires advance approval, unlike Americans entering Canada with a firearm who can get it done at the border crossing. You'll need to plan for 3-4 months for the approvals, unless that has changed in the last 8 or so years. When I checked into it, one needed to send the paperwork to an office in Washington DC and then wait, and wait.
You don't need any special permits to bring your firearm to Yukon, we're part of Canada after all. BUT if you shoot a bear without the required hunting permit, it can be onerous with the paperwork. Even if it is in perceived self-defence. It will definitely interrupt your trip for numerous days as you must take the dead animal to the authorities and file paperwork. The killing will be investigated, to ensure your life was threatened, not just because you saw an animal a few hundred yards away. The authorities keep the carcass too.
We live here and are often in the backcountry hiking and exploring. Bears often stroll by out cabin on their way to the creek, but have never caused a problem for us. Have never been threatened by a bear of any type. They tend to go the other way if they know you're around. Key is keeping a clean camp and not enticing them to join you. If you're on an ATV, it's unlikely you'll meet up with one because they'll avoid all the noise and ruckus.
In Yukon, be sure you are aware where First Nation settlement land is before heading off on a trail, as you cannot use this land without permission. There are different categories of settlement land, each having different types of restrictions. Also be sure you aren't riding your machine in park areas or private land. There are many tracts of private land in the bush without neighbours around. Also be sure you're up to speed on the laws applicable to off-road vehicles in Yukon.
