iPhone Medical app that retains information through an upgrade
Most iOS apps will transfer data to a new iOS device if the old device with the app data is Sync'd and backed-up to iTunes on a Mac or PC, and the new device is then restored from the backup of that old device on the same Mac or PC.

You might want to make a backup of your iTunes backups to be more safe.

If you don't want to restore from backup, but set up a device as new (blank), the you may need to find a medical web app where your info is stored online somewhere.

In the future, there may be some iCloud enabled apps (assuming they can somehow meet HIPPA requirements with your private medical info).