I have seen the term "strike out of the box" used on here and other sites, but I've never heard it used in the circle of hog hunters that I talk to.....Does it mean that the dog is winding while it is still in the box or on the atv/utv/boat/truck?
Yes that is when they wind a track from the box or vehicle or whatever, most commonly called rigging as someone has already explained.
Are the terms "Find Dog" and "Strike Dog" the same?
To me and the people I hunt with Strike dog means a dog that can find and start his own track, and find dog is a more general term for that but could also mean pretty much any type of dog that will try to find game or that's purpose is to find game. With us a strike dog could be open or silent as long as it is finding and starting its own track (striking its own track). I hope that helps.