just got ENC28J60 running with Arduino DUE with UIPEthernet-library
Not release or merged in branch master yet, code resides in branch 'due'. (You can download complete branch as zip-file
To install first unzip, then add the library by going to 'sketch'->'import library'->'add library', then navigate to the directory you have unzipped the library to and select directory 'UIPEthernet', then click 'open'. Afterwards you will find new examples (AdvancedChatServer, EchoServer, TcpClient, TcpServer, UdpClient and UdpServer) for UIPEthernet in 'file'->'examples'->'UIPEthernet'
I use Arduino 1.5.5, the ENC28J60 is connected to SPI on ICSP-header, CS is connected to pin 10.
Examples from UIPEthernet run without modifications, WebServer-example from Ethernet-lib runs by replacing include 'Ethernet.h' by 'UIPEthernet.h' (not removing the SPI.h, as this is required on DUE).