In your "My_objects" folder, make sure all the files for that mesh are there. Each mesh comes with between 3 and 4 files. (the .mesh, the .mat, and the .png). [excluding meshes that use existing mat files which must be typed in manually]
Everything should be in feralheart->media->objects->my_objects.
If you exported the files directly to that folder, they may have been placed in their own folder (since that's how i zip them, in separate folders). the game cannot recognize these new sub folders and the contents of them must be pulled back into "my_objects" for them to work properly.
The only other reason for them showing the basewhite mat is if you didn't unzip everything and only took out the .mesh file....which would just be crazy lol
I wish I could understand why a small group of people keep getting this problem.