Should I enable automatic security updates on a Ubuntu server? [closed]

It depends on what your machine does ultimately. Does it run mission critical applications that can not crash EVER? Probably not the best for auto updates then. Is it sitting on the egress point of your network? probably a good candidate.

It comes down to weighing security vs stability and finding what your acceptable compromises are. Most likely you're not going to be the target of a zero day but maybe you have very sensitive data that absolutely can't get out, it's a decision you have to make.

My suggestion is to design your network in such a way that the most security filtering is done on the edge of your network (blocking external applications, smart fire-walling, possible DMZ, etc.) and then treating the rest in a way that is most appropriate for your organization - whether that means nightly reboots with updates or weekly or automatic :)

Yes. You should enable these automatic updates. You are far more likely to have your system compromised by missing or delaying an update than for these updates to impact your running system negatively.