Not necessarily... frostbite comes from some combo of cold + dampness + wind/draft, but just walking on snow or ice by itself shouldn't be enough to cause frostbite. 20 degrees is no issue for chickens. -37 could be a problem, so having a draft free and dry roosting area and/or outdoor areas that offer some relief from winds and weather is important.
This ∆∆
And very well ventilated so moisture from the chickens breathing and pooping does not settle on the birds.
I do cover my hoop run but ONLY because it is on the north side of that coop so ices up and stays iced if I don't.

I made it so I can raise the sides or lower them and left a 6" gap next to the building for ventilation on the run.
The building has open soffits on the uphill and downhill sides.
There are 3 windows and even in winter I will open the South one during calm or sunny days.