Can you sign in to your Xbox one account on an Xbox 360?
You can sign into the same Xbox account, yes. Both Xbox 360 and Xbox One use the same Xbox account to log players into Live.
However, unless you own an Xbox 360 version of Minecraft, you will not be able to download the game, as Xbox One games are not compatible with Xbox 360. You won't even be able to access the games list from the store.