People seem to misunderstand what Push Notifications are
They're pushed from a server. In other words, data is stored on a server and the server goes "hey look, we need to send this to the phone."
Normal apps cannot "push" to themselves. The notification has to come from somewhere.
What you can do with the calendar app is setup reminders/alarms for each event. Have it set to display an alarm 5-10 minutes before an event. That's about as good as you're going to get. And that will only show you a dialog that you can see, and an alarm sound (should you set one). It won't display a badge on the calendar icon.