well, for a start you need to make sure the texture file is in PNG24 or DDS format.
X-Plane will not read JPG files. Also make sure the texture is in the same folder as the ac file (and also in the same folder as the OBJ file after export).
The texture dimensions must be multiples of 2 - I used 512 x 512 px on the one I included below. Don't make them smaller than 16 x16 px or bigger than 4096 x 4096 px though since this may cause issues.
The only changes I made to your box were applying my own texture and setting the faces to one sided (not much use for double sided unless you view the box from inside).
I also moved to box a bit along the Y axis to make it stand on ground level.
Then I exported to x-plane 8 format (File -> Export -> X-Plane 8 Object File).
copy the included png & wood_crate.obj into your custom scenery's 'objects' folder and try again. works here...
here's the modified object
peter