When you right click on a file in explorer, you can choose to Open with .. or Send To. You can add applications to your Send To.

Create a Shortcut to your application (right click the ProgramName.exe) and copy (or cut) the Shortcut to %windir%\Profiles\YourUserId\SendTo.

Now, when you right click on that file with a non-standard extension, you can Send To your application.