I wish I could post the instructions on how to make this box, but they are copyrighted. However, this is a great idea on wrapping a gift. I am sure origami cube instructions can be found by doing a search on Google or even Pinterest.
A friend of mine had a baby shower this last weekend and I have NO boy bags or gift wrapping material, I only have girl stuff. So I pulled out my 12x12" paper stash and found a coupled papers I liked and snagged my Origami book and found some instructions for a cube. Took a total of 8 minutes to make.
Then just tied a ribbon around it to give it a little extra niceness. If I was really good, I would have made a cute card, but I didn't. Shame.