Assuming it is a well-equilibriated system (which I'd say inside a slush machine is), then that temperature is zero celsius, or freezing point. As long as water and ice are in equilibrium, they will have a temperature of 0C (at sea level pressure). Eventually, there will come a point where the equilibrium is so heavily weighted to ice that the machine begins to seize up due to the sheer volume or solid material. Only once all of the water has solidified can the temperature go lower (in equilibrium).
Supercooled states can exist, but only in very still conditions free of ice. As soon as a crystal nucleates the ice will rapidly form (in like a second or two, look up supercooled water on youtube and even try making some yourself). Such a state would not exist in a slushie machine.
