It's supposed to get cold. They would both survive in the landscape, and outdoors is where they will survive as bonsai. The shed is unheated? They would need to be watered a little every 3 to 5 weeks in a shed. They don't use a lot of water, but there is some amount of evaporation when the winter humidity is at its normally low levels and the trees should never get bone dry. You can pile snow on the pots and that will both slow evaporation and provide some moisture for the plants' use. You also have to keep a dish of poison mouse tidbits there so they will not eat your trees. In the spring you will need to put the trees out in partial shade on warm days to get them used to direct sunlight, and back in protection overnight when it goes below freezing. No big problem, just part of the procedure.
Hardy trees are snowbird trees. They can be put outdoors in the landscape touching the soil, covered with leaves a couple inches higher than the pot in Oct. (Oak and Beech are best, but any will do). Then go to Florida for the winter and come back and uncover the pots about April 15th and all is well.