Problem Resolved!!! All it took was to plug the Brother printer into an outlet that was on a different circuit from the eMachines PC!
I discovered that the long beep when the PC shut down was not coming from the printer but rather the UPS "protecting" the eMachines unit. Both the UPS and the Emachines are plugged into the same circuit. Apparently the printer is leaking some kind of signal across to the UPS via the wiring which affects the UPS' logic circuits and causes a momentary power loss & alarm beep. Isolating the printer by plugging it into a different circuit eliminated the problem!!!!! Disconecting the PC from the UPS and plugging the UPS, stand alone, into the same new circuit as the printer resulted in it the same long beep and power shut down for the UPS when someone printed. I can't believe I spent all this time trying to find a software problem when all I needed was an extension cord from Home Depot. This has got to be one for the record books!!!
I discovered that the long beep when the PC shut down was not coming from the printer but rather the UPS "protecting" the eMachines unit. Both the UPS and the Emachines are plugged into the same circuit. Apparently the printer is leaking some kind of signal across to the UPS via the wiring which affects the UPS' logic circuits and causes a momentary power loss & alarm beep. Isolating the printer by plugging it into a different circuit eliminated the problem!!!!! Disconecting the PC from the UPS and plugging the UPS, stand alone, into the same new circuit as the printer resulted in it the same long beep and power shut down for the UPS when someone printed. I can't believe I spent all this time trying to find a software problem when all I needed was an extension cord from Home Depot. This has got to be one for the record books!!!