How can I prevent Cisco AnyConnect VPN from opening on startup?
Yes, I'm pretty sure you can stop apps from opening on startup even though it ignores Login Items. It is most likely in LaunchDaemons.

There are two locations for these, one is for Apple daemons only: 


  /System/Library/LaunchDaemons


The other is to include 3rd party processes:


  /Library/LaunchDaemons


You may also need to remove from:


  /Library/LaunchAgents


Just remove the files of the app you don't want to launch on startup and it shouldn't show up on launch anymore.