I can second the recommendation of ski goggles. For some reason I didn't have trouble with my eyes watering from the cold, but I did have trouble when snow was actually falling, and sometimes with flying road salt when it got windy.

I also recommend against tinted goggles. I wound up with a pair of those because they didn't look particularly dark, and I'd failed to find the clear goggles at the store I went to so I didn't have a good basis for comparison. And then the first time I tried to use them was during a snowstorm, at night, and then my headlight's casing cracked so that the battery kept falling out. They are, in fact, rather dark.