I want to travel in private jet from Auckland and please let me know that can I travel in it wit my pets

To where? Each nation will have its own health check requirements, health record requirement and quarantine requirements when a pet travels into its borders.

If coming into US, here are some rules
Dogs | Bringing an Animal into U.S. | Animal Importation | CDC

The issue is not flying. Since it's a private jet you can take take your pets on the plane. The issue is disembarking. You will need to go through customs if you land in a different country. And each country has their own laws about this.

Here's an example of what can happen: Johnny Depp's dogs escape Australian death threat | NOLA.com