Hiya,
Whilst you could give it a go, it might not be advisable. A refrigeration circuit is designed from the ground up to work within certain operating parameters including the temperature of the chiller compartment and the temperature of ambient environment. Dramatic variations to these parameters can cause abnormal operation, or even permanent damage.
I've seen this question pop up a few times, so I though I'd post an explanation for those who have an enquiring mind :)
Briefly explained, the refrigerant (gas) is compressed by the compressor and subsequently passed through the "condenser" (the fins on the back of the fridge) where heat is dissipated, and the refrigerant then condenses into a liquid. This liquid is forced through a small orifice (metering device) which sprays the liquid refrigerant into the "evaporator" (the cold plate within the chiller compartment. The sudden reduction in pressure causes the liquid refrigerant to evaporate back into gas and absorb heat, thus causing cooling effect within the chiller compartment. The gas then leaves the evaporator and returns to the compressor to begin the cycle again.
But the reason problems will occur when you attempt to lower the chiller compartment below its design temperature are as follows :
The metering device is extremely primitive in domestic refrigerators and passes a fixed amount of refrigerant, making "assumptions" that the chiller compartment will be within a certain temperature range. If you attempt to reduce the temperature below this expected range, the evaporator will end up flooded with liquid refrigerant (there is insufficient heat to be absorbed by the refrigerant, so the refrigerant fails to flash to a gas and thus remains as a liquid). This liquid will then return to the compressor.
When a compressor attempts to compress a liquid, very bad things happen - in the refrigeration industry, it is know as "liquid slugging", or "compressor slugging" and can ultimately lead to compressor failure.
You may get away with it or you may not, depending on the design specs of the fridge, but ultimately I'd advise using a freezer instead of a refrigerator if you want to achieve sub-zero temperatures.
Cheers,
T.