Negative voltage is just like positive voltage, except reverse polarity. Imagine putting a AA battery in backwards... that'd give negative voltage. Let's call terminal 1 the GND terminal, and terminal 2 where the positive voltage should be. Normally, terminal 1 is 0V, terminal 2 is 1.5V... when you connect a battery backwards, you could say terminal 1 is 1.5V and terminal 2 is 0V, except that we said terminal 1 is GND, which is what all voltages are referenced to. So, relative to GND, terminal 1 0V and terminal 2 -1.5V.
Anyway, negative voltage is completely independent of positive voltage, and things can still be powered, and current can still flow whether or not positive voltages are present or not. I haven't looked at how negative voltages are used on these boards, but it's common to use them as the negative voltage for amplifiers. That way they can amplify bipolar signals... if you used GND as the negative rail, any time the input signal went negative, it'd clip to GND (so you'd only get half the signal). I don't think there's any problem connecting a negative voltage to an amplifier if there's no input signal (since the rest of the board isn't running).
Some of the old boards used it as one of the voltages for RAMs and ROMs (they took +5, +12, and -5), though I don't think anything JAMMA would be using that. I've read that those could be damaged by running extended periods with a voltage missing.
DogP