There are many brands making proper shoes/boots for winter hiking. I would recommend taking a look at Haix, Lowa, Salomon Quest series, Lundhags and Meindl.

I prefer military boots, they have no branding and are typically designed and tested for decent longevity and comfort. My current ones are Bundeswehr surplus, I believe Meindl produced them. They cost about 60 euros or so, never been used.

Pick a size larger than you would usually do, unless they're explicitly designed as winter boots. In cold weather you'll want wool socks as well as regular socks and that requires some extra space.