:) I thought, "that doesn't sound like Barbara Kingsolver, he's moved on." When I Googled on this phrase, the first thing that came up was an actual page from Life Magazine from 1944. But then another hit showed me the place in "The Lacuna" where she is quoting it. I believe that in this context, "the country" does NOT mean a political entity, specifically the United States. I think it means "the country" in the sense of countryside, an area outside of cities or towns. I think it means "Japanese beetles have spread and now infest most of the middle Atlantic coast."

It's pretty simple, it just means "majority" or "most of" . For example: "The car has been broken down for a greater part of a year." means that the car has been broken down for most of a year. So, in your sentence, "the greater part of the entire country" just means "The majority of the country" or "Most of the country". This phrase is a little outdated and it would be rare to hear it actually spoken.