ENC28J60 is 5 V tolerant on data pins. This means it can be directly connected to 5 V SPI and reset pins of host MCU. The 5 V AVR MCUs read everything above 3 V as HIGH. This means there is no need to add level conversion between a 5 V AVR MCU and the ENC28J60.
The ENC8266 is powered by 3.3 V and can require 180 mA as top current draw while sending a packet over Ethernet cable. 3.3 V pins on common 5 V Arduino boards can't deliver this current. Uno and Mega R3 can supply max 150 mA on 3.3 V pin and the small Arduinos like Nano and Mini only around 50 mA. This means, for use with a 5 V Arduino use ENC28J60 modules with 5 V VCC option and power them from the 5 V pin.
There are at least 3 versions of the Nano Ethernet shield in shops. Black version is by Robotdyn, red and blue version are from different manufacturers. All these shields use the same schematics, but Robotdyn has a different PCB layout. The shields use logic gates (74HCT08) instead of a Tri-State Digital Buffer (74HC125) to convert the MISO line to 5 V . This creates problems. SPI device not selected by CS pin should release the MISO line by setting the output pin to HI-Z state. The enc28j60 of course does this, but the logic gate can't. It stays HIGH, blocking the bus. No other device can be used on SPI with this shield attached. (Many 5 V SD card adapters have the same problem with the level conversion on the MISO line.)
The new 3.3 V Nano boards could be damaged by this shield, while without the 5 V conversion the shield would be OK for a 3.3 V Nano. This applies to any 3.3 V MCU with this shield wired as module.
To remove the unnecessary conversion to 5 V, remove the 74HCT08 chip from the shield and connect the contact with a trace to pin 12 to the contact to enc28j60 MISO pin. The 74HCT08 has 4 logical AND channels. The shield uses two of them as buffers by tying both inputs together (1 and 1 is 1, 0 and 0 is 0). If you want to use the wake-on-lan function then connect the contact with the trace to pin 2 to the inputs of the channel. You can determine the right contact by following the trace or by measuring the continuity with a multimeter.
Modification of the common version:
Modification of the Robotdyn version: