I have been unable to find out if there is a remote wireless shutter release for the GE X600 digital camera. Is there one?
No, there is no way for a remote trigger to be hooked in. There are no ports or IR capability for a remote to work with. Even the USB port appears to be limited to file transfer (either to a PC or a Printer) so it is unlikely that a remote control system could be rigged up via the USB port either. The self timer is going to be your only option with that camera.
General Electric makes digital cameras?! I would be surprised if they actually make it and it isn't just a rebranded unit.
As far as your question, based on the specs sheet I would guess it does not have an IR(Infrared) port. It has HDMI, USB, and AV listed, but not IR at all. I've seen "hacks" that will trigger a camera from the audio port, but that won't be wireless. I doubt without some frankenstein setup you will be able to do this on the GE X600.
In other words, I would say that a wireless shutter release is not really an option with this unit. Your best bet is simply to use the self timer function.
Wireless shutter release is somewhat uncommon in most compact, point and shoot, and even prosumer superzoom cameras. It is more common in DSLR units. See What are some smaller digital cameras with remote shutter releases?