The Due, which is really a Atmel AT91SAM3X8E, only has 4 full hardware serial interfaces (though there is an additional UART which may work).
The first thing I'd strongly reccomend is to stop thinking of the board as an "Arduino". The Arduino tools just paper over the actual device. It's a AT91SAM3X8E dev board.
That said, the first place to start is to read the product page and datasheet(pdf).
Functionally, RS-422 looks like a differential asynchronous serial bus. Depending on the implementation, it can be half-duplex (e.g. data can only go one way at a time), or full-duplex (e.g. data can go both directions at the same time). This mode is determined by the hardware, as full-duplex takes more physical wires. Half duplex is one differential pair (and ground), full duplex requires two diff-pairs (and one ground).
For full-duplex on a non-bus topology (e.g. only two devices), the hardware required to interface the bus can be as simple as just sticking a differential line driver/receiver in between your MCU's USART and the other device.
For bus-topologies or half-duplex connexions, it gets more complex, as you need to be able to turn off the line-driver to allow the other device(s) to talk over the shared connections.
Also, one note is you'll see lots of "RS-422/RS-485" line drivers, etc... This is because the physical layer specifications for the two are the same, so a driver that works for RS-422 will generally work for RS-485, and vice versa. In fact, there are often devices that support both protocols.
Fortunately, RS-422 looks exactly like RS-232, simply with different physical signaling levels (and one is differential). As such converting RS232 to RS422 is as simple as just sticking in a converter. (RS485 is more involved).
This means you can probably use the 5th UART in the AT91SAM3X8E for your 5th channel without issue (or you could just use it for the RS232 connection anyways).
With regard to how to actually make your device talk RS-422, you will need a RS422 driver IC. Fortunately, there are lots of options. There are even options available in DIP for easy prototyping.
Breakout boards are harder to come by, as RS-422 is pretty unusual in anything hobbyist grade. I found one, but it looks pretty poorly designed (no ground terminal!). It would probably work fine for experimentation, though.
Assuming the ENC28J60 is otherwise correctly connected to run, you only need the connections you mention to a host to communicate with it over SPI. In theory, a host can do everything it needs to do by connecting to:
PWR and GND
CS (SPI chip select, driven by the host)
SCK (SPI clock, driven by the host)
MISO (chip to host data line)
MOSI (host to chip data line)
It can be handy for the host to also control the reset line, and there is a interrupt signal out of the chip if I remember right, but these connections are not strictly required. All management of the ENC28J60 can be done over the SPI. Using some of the additional lines is for convenience only.
Best Answer
Using the version 3.8 or higher of the Linux Kernel, you can set a Device Tree Overlay.
By this method, at runtime, you can incorporate the system device drivers.
Personally, I implemented the necessary hardware to operate with a RS-485 interface and Modbus protocol. This basically is the hardware for Bus transceiver and its associated components. When I wanted to start the operation, I could see that the ttyO4 port was not enabled.
I found the solution here, where the code is presented to compile an Overlay to enable the port ttyO4. Once implemented this, everything worked properly.
Attention
In Beaglebone board I used, the operating system installed was Ubuntu. When compiling the file with the Overlay, compilation tool threw an error concerning did not recognize the option "@". This is solved, and here's how to apply the patch for the Overlay compiler. Therefore, I recommend the Overlay compiler what you obtain source code instead of using a repository.