According to the Wiki the edmini has an ext3 data partition. That means it has a journalled filesystem, which means that after a power loss the filesystem can always be restored to a valid state.
Your filesystem should be able to survive a power fail. (More or less. When you have a 2nd powerfail while the fs is restoring to a valid state, it could get some damage, I have read somewhere). But that's not true for the files itself. If the box is writing data while you pull the plug, the filesystem will be valid after a reboot, but the file being written can contain garbage.
So it's recommended to shutdown the box properly.