Differential ADC where input goes below ground rail?
First, that question should be clearly answered in the datasheet for the device.  The absolute maximum section will tell you what voltage range the inputs can tolerate without damage to the part, and the operational spec will tell you what input voltage range it can sense.

Second, it is very unlikely that the A/D can sense voltages outside its power supply range.  Again though, read the datasheet for the definative answer.

Most likely you will need to put a resistor divider in front of your A/D.  That attenuates signals, so the signal level into the resistor divider can be larger than the native signal range the A/D can convert.  Note that the second resistor of the divider (the one your input signal is not attached to) doesn't have to go to ground.  In your case, you may want to connect it to the positive supply, or some other fixed voltage source.