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.