How can I stop an application from updating itself?
Maybe you can use an app such as Network Connections which has live capture capabilities to find the IP of the update server your app is accessing.

You can then add an entry to iptables to drop the connections to that IP.

For the changes to persist across reboots you'd need a kernel with init support. So you'd have to check that out as well and instal custom one if needed.

PS: I want to make clear that I have no real experience with actually trying any of it and that my answer is just put together from answers I found on other questions on Android Enthusiasts Stack Exchange. In particular, this one and this one.