Here's a resource for you.
http://www.nohvcc.org/noflash/index1.htm
The common term nowadays is Off Highway Vehicle (OHVs) as ATVs are traditionally 3 or 4 wheelers. Personally, I ride a motorcycle (2000 Honda CR 250) here in NJ. We are struggling with the same issues (as is every other state). There is a concerted effort by environmental groups to limit access to OHV operators and the environmental groups are far more organized in both getting media attention and organizing voters. Just look at the laws that have now limited access to snowmobilers in Yellowstone and other parks for evidence of it.
Enviros cite noise, emissions, and erosion problems as reasons for banning OHV use. As OHV manufacturers modernize their designs (and two strokes are phased out) emissions are becoming less of an issue but that argument will not go away. IMO, carving out trails with OHVs does not do significant damage to the environment but it's easy for those in opposition to them to take a politician out to some of the worst examples and convince them that it's the norm. The noise pollution argument can be mitigated with silencers and spark arrestors but until OHV enthusiasts show they're a political force by rallying votes, we will continue to see our interests ignored.
I hope to open a private OHV park in NJ at some point in the future but finding a site has been difficult.
As far as where people ride in NJ now, it is mostly either on private land (with and without landowner permission), on public property (in some cases this is legal and in others illegal), or in a park operated by a non-profit group in Southern NJ. The park is on state land.
Liability insurance isn't much of an issue as far as I know. If you are to ride in NJ on public land you need to register your vehicle but no insurance is required.
--Paul