Blender Game Engine
While waiting for EVEE, Clay Engine and other things that are currently under development, this should be somehow possible with the good old Blender Game Engine. Here's a possible result:
First step is to switch the Engine in the Info bar (usually is on the top of the screen).
Real time shadow
Add a Sun lamp to the scene and go to the Properties editor and open the Object Data tab (sun icon). You'll see options that differ from the ones exposed in Blender Internal. If you enable Shadow, you should immediately see the cast shadow in the viewport once you have set it to Material Shading mode.
Here's a summary of the fields you'll have to work with:
You can now tweak the parameters of the rendering of the shadow (color, resolution, algorithm,..) to fit your needs.
In my example I parented the light to the camera (notice the dotted line in the previous image) in order to make it move accordingly to the view and not fixed with the scene. So light always comes from top left regardless the point of view as it happens in 3D Builder.
For the whiteness of the materials, it's now just a matter of picking the right shade of white as diffuse color, bump up the intensity, and if you are not satisfied, also adding some Emission (called Emit in the material editor).