The green areas mean some sort of recreational / park area, but not always necessarily government land. Golf courses and privately owned park areas are often green as well.
You won't find forest service / state forest campgrounds by searching for "campgrounds" on google maps - the results that come up are those that pay to be listed.
The green areas on the map around your indentend location appear to belong to one or more National Forest. Check out this website:
http://www.fs.usda.gov/activity/psicc/recreation/camping-cabinsedit:
another great resource are the MVUM - motor vech. use maps
http://www.fs.usda.gov/detail/psicc/maps-pubs/?cid=stelprdb5177824
