Maybe you feel like crying is a negative thing and that you as a professional should be able to prevent that from happening. Or maybe you feel like you are responsible for making them cry. I'm not a therapist or planning on ever becoming one, but whenever I'm talking to someone about their life and they start crying, I immediately feel guilty as if their pain was my fault.

Maybe your lack of empathy is a way for you to keep boundaries. No one can hurt you if you aren't available for them to hurt.