in this case MPLAB can't help you. You need a EEPROM programmer/reader like Elnec, BKPrecision or else. There's some free stuff and easy to build like IC-Prog or Pony Prog with SI interface or JDM.

Free softwares:
Pony Prog: http://www.lancos.com/prog.html
IC-Prog:http://www.ic-prog.com/index1.htm

SI interface :
http://www.lancos.com/siprogsch.html

JDM interface :
http://www.ic-prog.com/index1.htm

OR, still use IC-Prog or Pony-Prog software to edit your EEPROM dump, use your PC to send the whole EEPROM dump to your PIC, then from your PIC to your EEPROM. In this case, some VB or else PC LANGUAGE knowledge are needed.