What OS are you running? If OS X or Linux, read on!
I'm not sure about automatically copying the highlighted results to the clipboard, but I do workaround this quite a bit as well.
The easiest way to accomplish this for me without leaving NetBeans is to simply open a built in terminal window through Window>Output>Terminal (in 7.2.1) - I then navigate to my project, and run the RegEx that I built in the Find feature with my tool of choice. In fact, I use the built in terminal for this type of quick stuff in NetBeans quite a bit. If running Linux, using clipboard tools like xsel (http://linux.die.net/man/1/xsel) in combination with a built in terminal emulator can allow for devising some nice workflow shortcuts within IDEs if you are more comfortable working/coding at a terminal. Note that built in terminal emulators like the one in NetBeans is likely not going to play nicely with cut/copy/paste using the mouse, for various reasons that I won't get in to here.
As far as a built in/extension based solution for something like this, it would be helpful! I am not aware of one.
Hope this workaround helps in the meantime.