Interesting question!  I'm not actually sure what your legal obligation would be, (not being an attorney,) but my gut feeling would be to go with the principle that he who uses the road should pay his fair share of the upkeep.  Both the U.S. and Maine Constitutions state that private property cannot be taken for public use without due process and just compensation.  According to Jordan v Canton, public use of a road without public maintenance will inevitably destroy the road; access is a property right attached to the land, and destruction of access amounts to a "taking," requiring due process and just compensation.  So one could argue that the public should not be using the road unless they contribute to its maintenance.  

Does the public lands property count as a parcel "benefited by" the road association's roads?  If so, that may mean they should be a member.  But who is the "they?"  I recall seeing somewhere that if a town owns a lot benefited by a road association, then the town owes their share (although I don't know if that's enforceable for a property held on a tax lien - but that's a side issue.)  So I would think that whoever administers the public lands should be notified that the access is maintained by a road association and not with public funds, and that therefore if they expect to have access via the association roads, they must become a member.  

From your description, it's not really clear if the public lands actually abut the road, or if it's necessary to cross private land to get from the road to the public land.  Unless there is a "great pond" in the public land, landowners are not obligated to let the public cross their private land.  If there IS a great pond, the public has a right to cross by foot to get to it, but only over unimproved land, which would not include the road.  

I wonder how the land was acquired by the public.  Was it given to them?  Or was it taken on a tax lien because someone got tired of paying taxes for a lot that had no access?  The term "caveat emptor" comes to mind.  What is the point of acquiring land for the public if there is no way to get to it?  

So I'm not sure if I've answered the question, or simply added more questions to it, but hopefully someone with more legal qualifications than I have can do better.