How can I install an exterior electrical box flush in stucco?
It sounds to me like the caulk is meant to prevent water from getting into the wall, not to prevent water from entering the electrical box. To seal the box, you'll want to use a weatherproof cover with a weatherproof gasket. 



If the box is in an area protected from the weather, you can use a cover that is weatherproof when nothing is plugged in.



However, if the box is not protected, you'll want to use a cover that is weatherproof even when things are plugged in.



In my opinion, it's a good idea to use an "in-use" cover for all outdoor receptacles