I have this exact same issue with drivers: it's a well known Linux kernel problem with Realtek ethernet adapters, see here and here. I'm guessing you installed the r8168 driver from Realtek by hand on top of the r8169 driver that comes with the Linux kernel. Note that these numbers are not version numbers, but product names. Some people say the r8168 driver from Realtek works better than the r8169 driver from the Linux kernel.
If I'm right, then you're installing a kernel module on top of the drivers installed by the linux-image-* package from apt. When you upgrade that script it rebuilds your kernel modules and doesn't know to re-install your custom driver. The problem isn't the driver so much as the whole kernel.
You could try pinning linux-image so that apt doesn't ever upgrade it, but then you lose kernel upgrades. Alternately you can reinstall the Realtek driver by hand after every kernel upgrade by re-running autorun.sh from the r8168 code. A third option, one of the links above has a solution that recommends blacklisting the r8169 module that comes with the kernel. However that may leave you with a kernel with no driver for your ethernet, I don't know enough about modules and ramdisks to advise you safely.