Thanks a LOT to the wonderful team of MakeHuman
that helped me to solve the problem (received the answer through Facebook):
I think what you run into here is that those programs you export to do not take the alpha channel of the texture into account. This ends up being a problem when you use the "high-poly" eyes, which have a transparent outer layer.
There are two ways to fix this.
The easy way: use the "low-poly" eyes instead. These do not have the transparent outer layer, so the problem should not arise.
The slightly harder way: Make sure the third part program properly links to the alpha channel of the eye texture.
For blender, the easiest way to achieve option two is to use the recommended import path (MPFB2) rather than exporting via an external format such as OBJ. For information on MPFB2, see
http://static.makehumancommunity.org/mpfb.html. With MPFB2 installed in blender, you should be able to simply open the makehuman save file for the character instead of first having to export the character.