VMWare support advised me to shut down all connection servers, then start one and see if the power on issue is resolved with one started. If it is resolved, boot the additional connection servers and see if the issue stays resolved or not. So I did that last night and my issue is resolved, in that the machines power back on after being shutdown from either inside the vm or from vcenter.
Note that when booting the connection server it takes a seemingly long time for the admin console to work, as the server has to start some services that are set to "manual" in addition to the services set to automatic startup.