Thanks, both of you, for your suggestions!
Here's the solution that I found based on what you told me:
Place a file called .gnomerc in /etc/skel with the following contents:
# Start the glipper clipboard tool
It took me a bit of digging to find out that I needed to do that (instead of the .xinitrc or .xsession file; putting the command there prevented the X session from starting). Basically, there's a file called /etc/X11/Xsession that sources the contents of /etc/X11/Xsession.d/, and one of the files in there, 55gnome-session_gnomerc, sources ~/.gnomerc if it exists. Somehow (I'm only a novice scripter, so I don't really understand it), the commands sourced through the above processes are then executed, so all one really has to do is make sure that the commands to be run when gnome starts end up in ~/.gnomerc.