Check that your wiring is as follows:
Atmega pin ENC28J60 module pin
50 4
51 5
52 6
53 7
The pins on the ENC are rated 3.3v but are 5v tolerant. However the voltage for powering the module is 3.3 volts.
Check you have a 3.3 volts on pin 9 and gnd of the power supply and Mega to pin 10 of module.
The Geetech manual shows the module running off VCC from the Arduino. THis would be fine if they had a regulator on the ENC module but the schematic for it shows no such voltage reduction. The Microchip ENC28J60 datasheet states that the ENC is a 3.3 volt device. Unless your Arduino is has A VCC of 3.3v you will damage the chip connecting pin 9 directly to VCC.
The Nuelectronics ENC28J60 shield schematic clearly shows the 3.3v onboard regulator and signal level shifters. Yours does not have these niceties but will still work if care is taken.
I can't recall if there is a 3.3v output on the Mega but do not use this if there is as it is usually only rated to 50 ma(from memory). Typically network modules pull ~100-250 ma.
Try connecting the module and PC to a ethernet hub/switch (instead of directly to each other) in case the crossover detection isn't working.
Also try cutting and pasting the code from the Geeetech "Manual source code.pdf" for a test sketch:
This is different to the Ping test code in the:
http://www.nuelectronics.com/download/projects/etherShield.zip