How to connect to device via SSH over direct ethernet connection

The link given by @steve is appropriate, but if you want to try something simple first just to see what the device does you can use dnsmasq, which is a simple dns and dhcp server. Install with sudo apt-get install dnsmasq. If this also enables and starts the server you need to stop and disable it.

If, say, your device is on your ethernet interface eth2, and you have done sudo ifconfig eth2 192.168.9.1 to set the interface ip address, then you can try:

sudo dnsmasq -d -C /dev/null --port=0 --domain=localdomain --interface=eth2 --dhcp-range=192.168.9.2,192.168.9.10,99h

which sets up a dhcp server in debug mode (-d), not as a daemon, with dns (port=0) able to offer addresses in the range .2 to .10, and to hold the same ones for 99hours.

If your device broadcasts a request to discover an address you should see it, and the reply (offer):

dnsmasq-dhcp: DHCPDISCOVER(eth2) 94:71:ac:ff:85:9d 
dnsmasq-dhcp: DHCPOFFER(eth2) 192.168.9.2 94:71:ac:ff:85:9d 

The numbers are the ip address assigned and the device's mac address. You may need to reset or power on the device if it has given up broadcasting. The device might reply to ping 192.168.9.2 and then you can try to ssh too. You can interrupt the dnsmasq once the address has been offered in this test and put the options in the standard dnsmasq config file and so on.

You may also usefully run sudo tcpdump -i eth2 to see what packets pass.