I remember reading somewhere that he does all the recordings using a 60's J bass
In all honesty, it doesnt matter what kind of bass you use.
Use stainless steel roundwound strings, always keep a fresh set on.
Play with a pick.
Bump the mids and treble a bit on the amp.
There are a few recording that really can't be Jazz basses, most of them do sound like he's using a Jazz, but album 5 sound like he's used a P to me. It's been a while since I've heard it, so I can't remember how much his sound changes on the ablum as a whole, but there a some where I'd be really surprised if it was a Jazz he was using. GGF sounds like he's using a Jazz though.
Can't say I agree with the stainless steel thing though, I never really see any punk bass players use them, although I'm sure they sound good for it. Not my kinda thing though, bit bright for my liking.
If your gonna buy cheap and live in the states, that's gonna be a much better bet then a Squire from what I've heard.
Basically the 3 generic punk sounds seem to be at the moment;
Jazz - quite alot of presence and high end usually, quite often used with alot of distortion - e.g. Rise Against, Bouncing Souls
Precision - bit deeper sounding in the mids then a jazz, susually used with a mid heavy EQ to exaggerate the already mid heavy sound, again usually overdiven, not quite as much as Jazzes seem to be overdriven nowadays though - e.g. Blink 182, NOFX (before Fat Mike started using a Danelectro)
Stingray - usually used for a even more exaggerated high end - e.g. Less Then Jake, latest Propagandhi
Those are just typical modern sounds, by no means all you can get from them, Klaus Fluoride gets the deepest tone I've heard in punk out of his Jazz. Don't feel there's any particular bass you have to use for punk, it's mostly you use them to get an aggressive punk sound. Just play about with your EQ and you'll often get what you need. Remember the amp you use is almost as important as the bass.
The conventions generally are one of those 3 basses, Ampeg SVT + 8x10, plenty of mids (and more commonly nowadays alot of treble too), new strings as often as possible, and playing hard with a pick. The punk bass sound has got real expensive, as seven has said, alot of people spend alot of money on sounding gritty nowadays.