The OBJ file format is mainly a geometry only shape based file format, made for the purpose of transferring basic 3D model data between applications
It is not a full featured save file format nor is it suited for transferring complex shaders or materials between applications, only basic material data is saved along with UV coordinates for texturing.
The tutorial you show seems to be using using image sequences as volumetric data to simulate volume density, which in itself is already an "exotic" type of material, and is very specific to the way Blender works internally. Your actual geometry is just a cube, that is why when you import you only get the cube.
You will have a hard time exporting it out of Blender in any way or shape, even less so using OBJ file format.
Now I have never used Expression Blend and I have no idea how it works, or what are it's rendering capabilities.
If it has volumetric rendering functionality you could try and recreate the volumetric setup in there using it's native material creation workflow.
Otherwise only solution I could image in Blender is possibly modeling a tight sequence of small planes so they form a roughly cubic volume. UV Unwrap the planes so they can use your image sequence as textures.
For performance and workflow reasons you may want to create a "texture atlas" from your image sequence, that means joining all you MRI sequence into one big tiled image, so you don't have to manually set up hundreds of materials/textures one by one, and so they don't ruing your performance loading hundreds of independent files.