You can test your JOGL based application using Sikuli which performs UI automation via image recognition techonolgy on screen-shots.
I am currently using Sikuli to functional-test a Java app that is predominantly based on the NASA Worldwind Java SDK (which is based on JOGL). Using the Sikuli Java API my test suite can recognise icons within the OpenGL canvas, click on them and also drag them around. Sikuli can also recognise and extract text from the canvas via OCR, however the performance of this seems to be a little hit-and-miss (depending on language, font, size and background colours behind the text).
I have done a lot of automated UI-Testing using other tools that work by introspecting the windowing toolkit (e.g. Swing, SWT, native Windows) and found that Sikuli runs much slower than those, however that is understandable given the amount of image-processing it needs to do behind the scenes. Also note that Sikuli currently requires your application to run in a Window (not in full screen mode).
Sikuli runs on both Windows and Linux. I'd recommend you try it out. I couldn't find any other tool capable of doing that level of functional-testing of an OpenGL based application.