Yeah, Dangling wires are never good. That wire is prob. a chasis ground. Most electronic devices have a signal ground and a chasis ground and sometimes a analog ground. Good rule of thumb is to keep all grounds seperate. It's not always posable. In your case I think the machine is shutting off because voltage surges are getting in. That's why it work better on the surge protector (they are regulated better). How is your power hooked up?
If your using a nornal 3 prong AC cord, you could hook the 3rd wire (green) of the AC to the that dangleing wire. I'm assuming that the dangleing wire is attached to some piece of metal or something. ?? If you really want to get sweet on it you might want to get a 5volt transient voltage supressor. Basicly, this device gets hooked up between 5v power and ground. and when a surge gets put into the system it will knock it out. For more info. Look up Transient votage suppressors. I know it's a lot to take in, but the bitch with PC's and microcontrollers is the nasty votage and current surges from poor power supplies. There is all kinds of ways to fix them but the easiest usually is the best. First I would hook the dangleing wire up to the AC earth ground (Green wire). Hope I didn't bable too much. If you have pictures of the wire I can tell you something better, but with out knowing much That's the best I could do. By the way I'm an electrical enginner, with a fetish for microcontrollers. and I only eat mother board when I'm out of romens soup.