Warning: rambling answer from an insomniac physics major contained within.
If the box is sealed, the helicopter and box form a closed system. Physically speaking, the question becomes not what the system weighs, but what the scale will measure.
If the helicopter is resting on the bottom of the box or hovering, the center of mass of the box + helicopter system is stationary. Thus, there is no net force and the force on the box from the scale is equal to the weight of the system. If the helicopter is descending, the center of mass is as well. The net force is downward (the gravitational force is greater than the upward force from the scale), and the scale will read low. The opposite is true if the helicopter is ascending.
The trucker with the canaries, however, will find his weight changing as they take off and land (unless his truck is sealed, and then the canaries won't be flying very long!).
I'll stop there because I should be studying for my thermo and electro exams instead of reading pprune
