Exporting to OBJ with textures not working properly I have a little bit of problem with exporting from Blender.

Basically doesn't matter what I do, it does not export the texture files properly. When I open the generated obj and mtl with text editor I cant really see anything about the texture picture. It's just says "usemtl Material.001" etc.. If I understand correctly the MTL file should contain the location of the texture file in some form.

Here is an example (very simple modell just to demonstrate)

Here is the generated OBJ, MTL and texture files. 

I double-triple superchecked in the OBJ exporting settings to check in "Write materials" also tried all the settings in Path mode. (No change). Texture file(s) are properly placed in the same directory as the OBJ and MTL, have a simple name, without space or any special character.

It's not the 2 programs incompatibility (I try to modell in Blender them use them in DAZ Studio) either because when I try to import the OBJ back to Blender it also misses the texture.

I use Blender 2.79 with no plugins or scripts. Tried to reinstall it, didn't help. (To be honest didn't expect it to but still tried :D)
You are using Cycles materials, the exporter doesn't support exporting textures from node based materials.
Since Cycles is acting as the active Render engine the materials you created are for the Cycles renderer, and hence use node based definition. The obj exporter can't read these materials nor find textures used in them if any.
Set Blender Internal as Active render engine, redefine your materials, and reuse desired textures.
Also see .fbx export why there are no materials or textures?