The system we have in the Scouts has essentially led to the creation of two very separate entities. In the higher levels, you have the Venture scouts, and the Explorers and what not, but for the most part, boys go to Boy Scout troops, which have their meetings, and camp in BSA camps. Girl Scouts go off and do...whatever it is Girl Scouts do.
But that's not how it is across the world. In many countries, you have the Scouts and Guides. While some remain gender exclusive, some groups essentially have troops of both boys and girls. If nothing else, perhaps due to lack of resources or space, major events (ie big hikes, extended camps, badge activities...career fairs, etc) will often be held in conjunction with "Brother" and "Sister" troops.
As a Boy Scout, I was myself pretty skeptical of Girl Scouts until we had a troop of French-Canadian Girl Scouts sharing our campsite for a week... Good God they were beautiful...
However, I should add that, again, in the upper levels, girls, or should I say women, are certainly present in the Boy Scouts. Girls can join the off-shoots of the main BSA group. When I worked at a Scout camp for a summer, we had a large number of female counselors...I think at least one at the major camp sections, all the way up to one of the sub-camp manager/directors. On top of all that, mothers of Scouts are usually involved in the troop, with many troops having female Leaders. I don't know, or think that's the case with the Girl Scouts...but I could be wrong.
That said, this specific case is less about the distinction between Boy Scouts and Girl Scouts, and more about the issue of where do transgender kids fit in to the whole thing. I wish I could say he'd be welcomed in the Boy Scouts, but I just don't think that's the case. Even if you find a "progressive" troop, chances are that somebody will complain and then like the issue of homosexuality, or religion, things could get out of hand.