Well, you have your answer:

Q. Is it possible to use a negative voltage on the A/D input for the pic12f675 ?

A. Yes is it possible (careful what you ask for).

Is it recommended? -> No it's not. You could damage the chip and besides your measurement will not go under Vss (GND).

You could sense the signal even without a negative supply, but here are the guidelines (if you want to operate from positive supply only):

1) You need the supply rail (Vdd) to extend somewhat above your signal range (i.e., if your signal can swing from +3V to -3V your supply needs to be +6V with some margin). Why is this?
2) To measure this you need to "reference" the measurement. For this you need an OPAMP.