I usually turn to the Principle of Explosion to explain why one should have core axioms in their ethics, (specifically non-contradictory axioms). If some principle you use in deciding what is or is not ethical creates a contradiction, you can justify any action on the basis of that contradiction. If the axioms aren't explicit, the chance of a hidden contradiction is higher. The idea that every action could be ethically justified is something that very few people will accept, so explaining this usually helps.
I try to understand that thinking this way is odd to a lot of people and that they may not have explicit axioms, and present the idea as "something to think about." I think this also helps me to deal with people not having explicit rules that they follow, since it A) helps me cut off the rhetorical track of "Well, I don't need principles" by extending the olive branch to the other person; and B) reminds me that many people haven't even tried to think about what grounds their ethics, much less what grounds what grounds their ethics.
I usually use the term "rule" or "principle" as opposed to "axiom," merely for the purpose of communication: most people will accept that there are core ethical rules or core ethical principles, but they may have never even used the word "axiom" before and be hesitant on that basis alone.
Usually, I don't get offended at things that people say to me, because I can see at what points in their argument we differ, and what sort of counterargument I could make to that. I can't get mad at people for having beliefs I think are wrong, since I myself regularly have beliefs that I later realize were wrong. I can't get mad at the idea, either, since either it's a thing that's right, or wrong, and if it's wrong, I have the power to say why. And if it turns out I'm wrong, so be it, I'll adopt new, right beliefs. And so I never got offended about anything.
Until one day.
One day, I encountered a belief that should have been easy to refute. Or, rather, easy to dissect, and see whether there was anything wrong with it, and if there was, formulate a counterargument. But for seemingly no reason at all, it frustrated me to great, great, lengths. My experience was as follows:
I was asking the opinion of a socially progressive friend on what they feel are the founding axioms of social justice, because I was having trouble thinking of them on my own. (They can be derived from any set of fundamental axioms that govern morality, but I wanted something that you could specifically use to describe who is being oppressed, and why.) They seemed to be having trouble understanding what I was saying, and it was hard to get an opinion out of them. They also got angry at me for dismissing Tumblr as a legitmate source of social justice. But eventually we got to the heart of the matter, and I discovered a basic disconnecf between us: they asked, "Wait, you're seriously applying a math thing to social justice?" And I pondered that for a moment and explained that it isn't restricted to math at all, and an axiom in this context can be any belief that you use to base your beliefs on. However, then the true problem came to light (after a comparison of me to misguided 18th-century philosophes): "Sorry if it offends you, I just don't think in general that you should apply this stuff to society. Like... no."
And that did it. For the rest of the day, I wreaked physical havoc, and emotionally alienated everyone I interacted with. I even seriously contemplated suicide. I wasn't angry at my friend in particular for having said that. For the first time, I was angry at an idea: that belief systems about certain things should not be internally consistent, should not follow logical rules. It was extremely difficult to construct an argument against, because all of my arguments had logically consistent bases, and were thus invalid in its face.
I'm glad that I encountered that belief, though, like all beliefs, since I was able to solve it in the end, and make peace with it. I came to the following conclusions: