You can't turn them off, exactly. You can reduce the likelihood of them occurring by opening your common\static_modifiers.txt file, scroll down to the "Global Static Modifiers" section, and add
monthly_character_loyalty = 1 under the base_values section. This will make all characters stay loyal all the time, but IMHO this removes one of the most interesting aspects of EUR.
What civil war problems are you having, exactly? If you're playing a republic, have a look at the Unofficial Strategy Guide linked in my sig. If you're playing anything else, just make sure your characters stay loyal by helping them out with their ambitions, etc. They aren't hard to avoid once you get the hang of things
