The id does have rights to the share.
We have other .exes that run fine. The only difference is that this one was written in VB and the others in C++ with command line parameters.
I did move the .exe off of the network drive and placed it directly on the SQL Server HD and now it works.
I even tried shortening the name of the exe on the network but that didn't seem to work either.
Anyway, it works now that I moved it, but thank you to everyone for the ideas.