This is old, but it still came across my google search. In short, you want components and not bodies. You can right click and export individual Components (as many file types), but not bodies. It seems with bodies a lot of people are saving them as stl's and then converting them, I assume because they don't know Fusion 360 that well; and the way it handles "assemblies" is quite different than other software I've used.
If you use the top menu to export as the other toxic human being commented before me, you'll export every body and component in the project.
So if you want to export individual parts, each part needs to be its own "component" not a separate "body." I still get hung up on this as I only use Fusion 360 once in a while for work, and recently created a project with different bodies instead of components myself.