I couldn’t get the network working correctly either, but then I stumbled on to this Arch Linux wiki page about the ESPRESSOBin. The “Default system configuration” section has a useful hint on how networking on the ESPRESSOBin works.
The ethernet port nearest the USB 3.0 port is the primary ethernet port, and is instantiated as the wan interface for connecting to the internet.
This is set up via wan.network. The eth0.network file is required to bring up the link on the eth0 interface to bring up all three ports.
That gave me enough to whip up a /etc/config/network file with a working wan interface.
config 'interface' 'eth0'
option 'ifname' 'eth0'
config 'interface' 'wan'
option 'proto' 'dhcp'
option 'ifname' 'wan'
