Can I put it outside?
You can, but it’s likely that pests like fungus gnats will make a home of it and get to your mushrooms first. The larvae stage of the gnat lives inside the farm eating the mycelium and weakening it. They will eventually eat their way through the cap and fly away to lay more eggs.
If you see little critters flying around your shiitake-farm, invest in some yellow sticky fly traps sold in garden centers.