If I right click on a program like REX or even FSX for that matter that is on my ( C ) Drive , and send it over to My External Hard Drive ( J ). Then delete the program that was on  My ( C ) Drive originaly just to free up space. Would that cause any problems ?

It shouldn't... Better get some more inputs before your try it though... You can test it by moving to your external without deleting the original... And trying it from there, but again don't delete the original until you get more input from other members.

I would be wary of trying that. If you're right clicking the shortcut for the program, that's all that will be sent to the external hard drive. Also you have registry entries related to the location of the program which I don't believe will be updated if moving a program by this method. This would mean installing addons would not work properly because they wouldn't find the proper program installation location. Before you try this, let some more experienced folks lend their expertise.