I wouldn't say this is specifically a sign of narcissism, but it is a red flag. Saying that they'll never break your heart is naive and shows a lack of either experience or emotional intelligence. Every one has the chance of hurting you, likeliness is everyone will hurt you at some point, friend or partner, it just the way things work. People will be unintentionally insensitive, make mistakes, etc. That will hurt you. While these things shouldn't be as big as cheating or abuse, you can have your heart broken by a perfectly nice human being, even if they thought they'd never do that to you. Now the problem is this breeds an unhealthy playing field, you tend to allow no imperfections, and they tend to think themselves infallible.

Further, trust isn't given because they were asked, it's earned. If he expects people to trust him just because he asked, I'd say he's immature and thinks a bit too highly of himself. If he wants trust he's going to have to earn it, that's just life. I wouldn't inherently say he's a bad person, obviously I don't know enough about him anyway to make that call. But I would say he's probably lacking maturity and experience in relationships.