Johnny cries at the church in Windrixville while speaking to Ponyboy about having killed Bob. This shows that he is sensitive, not a cold-blooded murderer, because he truly feels bad for his actions. If there were another way to have stopped the Socials, I think he would have choosen it.
Johnny is also sensitive because of the way his family treats him. He, unlike Dally, can not stand the fact that his parents do not care about him. He is not "hardened". That is why it hurts him so much to learn that his parents did not ask about him when he killed Bob and when Ponyboy said he was unwanted at home (when they were at the movies).
Johnny is terribly abused by his parents and statistics indicate that this would not make him sensitive, but it might make him want to be tough. This lack of love from his parents does affect Johnny. One of the first things he asks when he comes to is "Did my parents ask about me?" And his mother upsets him so much at the hospital that they have to make her leave.
The sensitivity we see in Johnny was probably a result of his genetic make-up. We see many children that are just born more sensitive than other. However, your question was how is Johnny sensitive, not why was Johnny sensitive. I am unsure what you might be really wanting to know. His demonstrations of sensitivity during the novel were numerous. The fact that he was so polite and nice to the girls at the drive-in, his ability to cry, his saving the children in the church, and his discussions with Ponyboy about poetry all indicate a sensitivity within him that does not appear in other characters.
"While Pony reads from "Gone with the Wind" about Southern gentlemen riding into certain death, Johnny sees Dally. And when Pony recites Robert Frost Johnny understands the meaning of the poem. They have to stay gold, stay young, and stay true to themselves. It is this message that Johnny sends to Pony in his final letter and the one Pony is left to struggle with."
When is he crying?
Well, it could be that his mother ignored him that's one.
But is there other reason?