Depends on your definition of "High Quality". There are English built guns which are not particularly High Grade but most are pretty well built. I have a J P Clabrough 12ga which is considered one of the lower grade guns but it is very well built & extremely serviceable. It does have a bit of game scene engraving, but is their entry level model. An I Hollis muzzle loader is a very plain gun but again well built. I also have a W&C Scott pinfire which I would refer to as a High Quality gun though it is actually not as "Sturdy" as the others. Breech loaders were still somewhat in their infancy when it was built.
Greener infers that there were a good many Shoddily built guns built in England, primarily for export to their colonies, of which we were not one of by this point in time so few of those show up in this country.