I'm able to import an object into Blender which was created in MagicaVoxel. The object keeps the desired texture in Blender and looks like this in Edit Mode.
The result should be useable in Unity as well.
Other solutions from this site like How to create a "Rounded Grids" effect like in MagicaVoxel sadly don't work.
I've tried to work with Normal Maps and UV Mapping but these approaches apparently didn't work due to the shape of the object and due to the fact that it comes with material.