Hopefully some helpful clarification. This applies to both BE and AE:
Out of the box is NOT commomly used as an expression in a figurative sense. The words are meant literally and require explanation:
This machine requires no assembly. It comes preassembled and is ready to use right out of the box.
It sounds very slangy to attempt to use the four words as a hyphenated adjective before a noun, although understandable:
That was an out-of-the-box cake, wasn't it? Better: That cake was made from a box mix, wasn't it?
He didn't bake the cake from scratch, but rather from a box mix.
Outside the box, on the other hand, is a currently popular expression that, as far as I can recall, is only used with the verb think.
It means to engage in inventive thinking, whether naturally or by intention, to have the capacity to think in a creative, unconventional way, which can result in novel solutions, inventions, art, etc.
Some people think in a very linear way, whereas others seem to have a special ability to really think outside the box.
The expression is too cumbersome to use as a hyphenated adjective before a noun. It sounds much better placed after the noun:
John is a wonderful asset in our creative team because he often finds solutions that are really outside the box. (or: highly inventive solutions)