What kind of cooler should I use in freezing temperatures?

You want a rotomolded cooler (like Yeti, Pelican, Engel, or ORCA), not a cheaper injection-molded one. Regular hard coolers use thin plastic that gets brittle below freezing and can crack, while rotomolded coolers are made from thicker, tougher polyethylene that handles cold without shattering. Just leave a little room for ice expansion in subzero weather.