Hi -
Somethings I would check right off the bat, look at the computer where the mesh looks corrupted and see if the Engine Scalability settings have been altered and if so return them to Epic and see if the mesh corrects itself. Also delete the Intermediate, Saved folders before you open the project on the second computer, this will force a shader recompile and hopefully remove any corrected temporary Shader Files.
Finally if none of those ideas work, can you let me know how you are viewing the vertex colors from the pictures above, is it directly from the Vertex Painting Preview or do you have a Vertex Color based Material applied to the mesh and if it is a Material can you post a screenshot of it? Finally and just to be thorough can you upload a copy of the DXDIAGs from each CPU?
Thank You
Eric Ketchum