How can I stop unattended-upgrades from rebooting the machine?

Verify that in the configuration file: /etc/apt/apt.conf.d/50unattended-upgrades you have set:

Unattended-Upgrade::Automatic-Reboot "false";

This will prevent the software to issue reboot also if needed by the upgrades.

Documentation

To disable automatic upgrades altogether, just remove the package. Try:

sudo apt-get remove unattended-upgrades

See Ubuntu docs for more info. However, if you want to keep the package and configure its behaviour, see this question's solution How do I enable automatic updates?