The lower limit is ~0.3 volts. One way you can protect the input it to put a resistor in series with the input.
I would suggest a 5k resistor then a 0.1µF to ground right near the pin.
The 5K will limit the current so you can go approximately 5v below ground and not have enough current to damage your DUE.
The 0.1µF capacitor has nothing to do with negative voltage but will make your analog readings less likely to pickup noise, giving your better readings.