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.
In normal serial communications links, you actually need two wires - one is the signal and one is the ground. The ground stays at zero volts and the signal wire swings between 2 voltages that represent data bits 0 and 1 respectively (in RS232, the signal swings between +25 volt and -25 volts).
Using voltage to signal is not great for long wires going though noisy environments because the wires act like antennas and the voltage difference can get lost in the noise.
RS422 uses current direction rather than voltage to transmit bits. Current going one way is a 1, reversed the other way is a 0. This is much more resistant to noise.
The RX+ and RX- are the two wires that that current for received data. You can imaging sending data down this wire using a battery as a transmitter. Connect the battery one way to send a 1, or flip it the other way to send a 0. Two wires, but they carry one bit of data. You can connect this one bit of data to a single serial input (using the the correct drivers to turn the current direction on thew wires in a voltage to show the pin).
Same goes for the TX+ and TX- wires. They carry a single bit. The driver will take a single bit of data represented as a voltage on your sereial send pin, and convert it to a current flowing one way or the other on the two TX wires.
ADVICE: Unless you are making lots of these, don't mess around with driver chips (unless you want to!). You can cheaply get prepackaged current loop dongles that have screw terminals and are very easy to use. They use the same chips you are looking at, but do all save you the hassles of figuring it all out. If you get one that uses TTL level signals, you can connect it directly to your Ardunio and treat it as just a regular serial connection. Here is an example of one (I've never used this one but looks as good as any other)...
http://www.amazon.com/INBOARD-RS422-DRIVER-ET-MINI-SHIPPING/dp/B00EXUGRUK/ref=sr_1_1?ie=UTF8&qid=1392255264&sr=8-1&keywords=ttl+rs422
Best Answer
An incremental encoder, so you have two signal lines plus maybe an index pulse, at a minimum.
You should be able to use RS422 or RS485 drivers (the latter can be constantly enabled) and receivers to send a differential signal over a few pairs. If you use shielded twisted pairs, properly terminated, it should be more than okay. If the EMI environment is hostile, consider isolation.
You could use inexpensive shielded Ethernet cable if that suits your other requirements.