If by "quad maps" you are referring to USGS quadrangle topo maps, they show topography and geographic features, but they don't address ownership or legal status. They are compiled primarily from aerial photography, with field checking to maintain accuracy.
All the line on the map means is that there is a geographic feature there which has seen enough traffic to make a semi-permanent route that the cartographer can see in the aerial photos. The USCGS does not check to determine if such roads and/or trails are or are not public rights-of-way. Anything that is not considered a public right-of-way by the legal authorities in your township or county is private property, and the owner can deny passage at any time.
In some cases, there may be a public ROW bounded on both sides by private property, and you can drive the road (even if it's a dirt trail) but the entire length will have "No Trespassing" signs along both sides. In such a situation, you can't go off the legal ROW but you can traverse the ROW.