Horses do have great ability to adapt to cold weather, Laura. Breeds that originate from cold climates adapt much better than others, but all horses develop a thick coat and a good layer of fat. I am always amazed to see my horses transform from sleek, thin-coated beauties to puffy, round creatures in the cold of a Kansas winter!
However, it is important that horses have access to shelter. Three-sided barns, facing south, are good pasture tools. The ground in the barn should be clean and dry, which means it should be higher than the ground around it. There should be access to water with an automatic waterer or a heated water source, and plenty of accessible hay.
Whether horses prefer to be outsideI think their actual preference is the more enriching environment. A singly housed horse who cannot touch another horse and doesn’t have access to enrichment toys would likely be happier if he could go outside, where he would have a range of possible behaviors from which to choose.