"Non-apologizers tend to walk on a tightrope of defensiveness above a huge canyon of low self-esteem - they just can't listen to anything that's going to set them off balance." (Harriet Lerner)
One way you can tell you're dealing with an egomaniac is how they react defensively whenever they made a mistake.
"LOOK, I'M NOT PERFECT, OKAY?!?"
The first move in response should always be to pause. Give your best brain a chance to lower your heart rate. Then,
...if you're ready to challenge them, here's a great response: "Umm, dude,...no one's confused about that." ...if you're wanting to connect: "Are you feeling attacked? If so, how?" ...if you're a leader charged with helping this person grow, try this: "Is this how you normally react when you make a mistake? We all make them, but we can only learn from our mistakes if we don't get so defensively reactive."
Perfectionism is a disease that can plague any one of us, but it's especially attractive to the egomaniac. If you want to create any change in your relationship with them, then either confront them with the truth (no one's perfect, and no one thinks you are), or connect with them about the pressure they feel and invite them to see it differently.
(If they're open to changing, then your challenge or connection may start something new. If they're truly an egomaniac, then...good luck.)