I don't think there is a way to do what you are asking directly with the mouse software. There are no options in the software to allow a button to start an application directly.
However, you can bind a mouse button to any of the function keys (F1 thru F12). If you could find a program that would allow you to bind the application to these keys, then yes, you would be able to start the program with a single mouse button.
Hope this helps....