A barefoot horse gets better traction than shod horses in snow. First, being bare, they have a better feel of where their feet are and adjust when needed. Second, barefeet by design just get better traction, they kind of dig in and the frog helps grip, with a reg shoe the ground surface is just an incomplete ring of metal that is too broad to really cut in as the foot breaks over and it suspends the foot above the ground, where it can't feel as well.
Shoes also help the snow pack in and ball up a lot more than a bare foot. Barefeet even get better traction on concrete. Think of it, the hoof expands and contracts with each step, and the frog is almost like rubber-think shock absorbing tire (smoother ride, better traction)compared to a metal shoe (steel rims on your car?).Shoes, by the way really inhibit the hoof expansion, which is what helps in blood circulation inside the hoof.
If snow is really sticky, it can still ball up, even in bare hooves, so you can coat the sole with vaseline or spray Pam (cooking spray) on to help the snow not stick. You would have to do this every day while the snow is sticking, so best to apply the fresh coat just before you ride.
There are boots you can buy that you can attatch studs to or are designed for mud/snow conditions, but aren't suitable for regular footing, but if you plan to work your horse in bad footing, these would be a good idea.
In the mean time, just keep picking those feet out. Esp when it's muddy. Even if the just plop the foot back in the mud immediately, any exposure to air will be benificial in preventing thrush.