UniFi security gateway keeps dropping the internet connection

Before the logs, does it come back if you just unplug and replug the WAN cable instead of rebooting the whole USG? If yes, I'd bet on the DHCP lease from the ISP not renewing properly, which is a really common one on that box. For logs: SSH into the USG with the device credentials from the controller settings and run show log, or tail /var/log/messages. Look at the time it dropped for dhcp or WAN lines. Easier long term is turning on remote syslog in the controller and pointing it at a PC, because the USG's own log doesn't survive a reboot, so if you reboot first you lose the evidence. Also try switching off IPS/threat management for a week if you've got it on. It works the little USG very hard.