A simple script might look like this:
Hello, this is Sue from XXX Animal Hospital. I'm calling today because we are updating our records, and found that "Snoopy" has not been in since "July 2014", and our records indicate he/she is overdue for both an annual wellness exam and important immunizations.
I'm happy to answer any questions and would like to schedule an appointment to get "Snoopy" in to see one of our veterinarians in the near future. What time would be good for you to come in to the practice?
The person making the calls should make this their own - not just read it - but the message needs to be clear. You are overdue and need to setup an appointment. If they decline, you can make a note in the records and move on.
For leaving messages, something like the above will also work. "Please call back to set up a convenient appointment time for Snoopy" - it is important to mention the pet's name once or twice.
Another comment would be that it is important to address any minor issues with Snoopy before they become a major issue.