There are two main types of mice in Canada, southern Ontario especially as we have a lot of house mice. House mice are not native to this country or even this continent. They are from Europe and came here on ships along with rats. They are unable to live in the wild and must make a home in our structures in order to survive. They do not know how to survive the cold and will simply die. If they are in a human home, however, they will not die from the winter as they will have access to heat. This is important regarding getting rid of an infestation. Turning the heat off will not hurt them as they are already away from the wind, as long as they stay together you can make it as cold as you want, they will not die.
Wild mice are a different story. Like most wild animals that stay away from humans, they do not really know what a house is. This is demonstrated by the fact that they do not know your home contains food or Is even occupied by anything. They only feel the warmth coming off of it in the winter. Without our heated homes, these mice would indeed have to burrow into the snow in order to survive and thankfully are very good at it. Unlike house mice, they have all of their natural features intact. They can jump great distances, climb too structures from trees to houses and they are much less likely to access your garbage but much more likely to access your bird feeders. They have a tendency to infest car engines which can cause severe damage, just like in the walls of a home. They can chew wires and stop your car from starting. Repairs like this can cost thousands. How would you deal with mice outside? The same way you deal with rats. Poison on the outside of the house. But what about the other animals? You can keep them out of the bait by putting the bait in mouse-only tamperproof bait stations. They can be locked with a key and are very hard to break open.
If you have attic mice you can treat them yourself. Unlike house mice, their population never grows very high and they only occupy the space during the coldest months. However, even in that short time, the damage to your attic can be extreme. From chewing wires and starting a fire in your attic to defecating and urinating in your attic ruining your insulation permanently. If you have mice on your property, in your car or in your attic you should call a professional. They will be able to inspect and see the full reach of the mice if they are wild or house mice and how to best treat them where ever they are. They can also seal openings as entryways in structures to prevent mice from entering your home or attic.