Does DayBack Sync to Google Calendar
DayBack shows your Google Calendar items and your Salesforce, FileMaker, or Basecamp items in the same view so that you have one place from which to make your scheduling decisions. You can create and edit both Google and Salesforce, FileMaker, Basecamp, etc. items from within DayBack, and you can do this with not only your Google calendars but any that have been shared with you.
So DayBack does not sync or publish your other events to Google Calendar.
If you need to publish and share events to users outside of Salesforce, you can use the sharing feature in DayBack to create URLs for any view in your calendar. This is great for sending customers their project timeline, or sharing schedules with employees or contractors. Details and a short movie of this in action are here: https://dayback.com/publish-share-schedule/
Companion Apps that will Publish to Google Calendar
DayBack does not sync or push your Salesforce events to Google Calendar, though there are a few App Exchange apps that will do this alongside DayBack...
Appirio will sync events to Google on the hour, but they can not sync custom objects.
Most of the calendar sync plugins for Salesforce sync your whole events table and don't allow much filtering... but, I've found what could be a great solution in that it would let you sync on conditions like fields in the event or staff record. The app is called Workato and here is an example "recipe" describing a sync: example. If you click on any part of the recipe you'll see the field mapping and filtering available. I've chatted with their support--they seem very responsive--and they get 5-star reviews on App Exchange.