Are you using PhoneGap to create your app? If you are then two things:
You can store the actual app.apk file in the internal memory (or SD Card) and therefore reinstall it whenever and wherever you like.
You can save the database anywhere on your internal memory (or SD Card) and it won’t be deleted if/when the app is uninstalled.
To be honest I can’t even begin to imagine why your app would continuously be uninstalled if it’s actually required, but there are apps out there that can prevent other apps from being uninstalled. I don’t know if I’m allowed to advertise other products here, but if you search online you’ll come across parental apps that allow you to restrict certain aspects of the phone unless you have a password. I use one on my son’s phone and I can restrict just about everything on it, including the ability to uninstall apps.
Hope this helps.