1. Intimacy - not relationships. He's fine with things like dating, hanging out, making small talk, - will and can do so with both sexes and is entirely capable of being romantic or wooing someone when he has a reason to (and he has done so to manipulate). Kissing, touching even platonic touching like a hug on occasion, sex and the possibility of sex, and the intimacy of marriage (sharing his life with a person) bothers him a great deal.
2. His masculinity. Though in that case it is a defensive reaction to constant attacks against it. His curse and the situations that he gets in has put a great strain on his views on what it is to be a man.
Ranma accepting challenges and the reversal jewel thing really has nothing at all to do with insecurity in my opinion. That has to do with the fact that Ranma is extremely competitive.
Like the shouting of "I'm a guy!" or the assertion of "I'm the best" even when he gets beaten he doesn't admit it as a defeat even while training for the Rematch!
Is that an anime thing or just fanon? Ranma doesn't assert he is the best in the manga, he only shouts that he is a guy when someone wants to put him in a situation that is clearly socially for girls and doesn't do so often at all, and can admit defeat but doesn't dwell on it instead attempts to improve or find a way to win which is because he is competitive (he also has turned down a challenge in the manga until he got pissed off and decided that he wanted to teach the opponent a lesson).