Not much on power consumption for these ethernet jacks, but I did find this:
https://groups.google.com/forum/#!topic ... PGrRTJjReA
They quote power consumption from 500ma - 1.5A! OUCH! I had seen power consumption for the zero to be quoted around 200-300ma, pushing a possible total to as high as 1.8A, so I am willing to bet that power supply is a big issue for many.
After looking over my syslog file, I found the solution posted here, which is working pretty well for now:
viewtopic.php?f=91&t=122601
example of line in /var/log/syslog:
Oct 19 14:00:44 raspberrypi1 rsyslogd-2007: action 'action 17' suspended, next retry is Wed Oct 19 14:02:14 2016 [try http://www.rsyslog.com/e/2007 ]
Come to think of it, I did first run the Zero with a monitor and keyboard, and did not seem to have any difficulties, so that definitely points to the ethernet as the issue.