Hi Mika2013,
From the error message” Unable to establish a connection to the authentication service. Contact Technical Support.”, I suspect the issue may be related to the DNS. To narrow down the issue, for example, you can clear the DNS cache in the AD DNS service, and restart the internal DNS service to see if there is any improvement (if you have deployed the AD DNS service).
As for the issue, here are my suggestions.
1. Try a different Office 365 admin account such as admin@domain.onmicrosoft.com and enterprise admin account to set up Dirsync.
2. Make sure the correct password is being used for the "onmicrosoft.com" global administrator account.
3. Uninstall the current version of Dirsync, and install the latest version of Dirsync via http://go.microsoft.com/fwlink/?LinkID=278924
Thanks,
Johnny Zhang