In The Catcher in the Rye by Salinger, what is the illness or disorder that Holden suffers from that makes him push away from social interactions? What are three quotes that support that?

In the story, Holden appears to suffer from clinical depression; this disorder is characterized by a lack of interest in life and a tendency towards low self-esteem. You will notice that Holden talks a lot about being depressed in the short novel. He is also saddened by seemingly mundane things, another symptom of clinical depression. Sometimes, Holden is also depressed when he thinks about the kind of person he is.
In Chapter 13, he talks about being depressed because of his cowardice. He thinks that, if he had the ability to beat up someone when he felt like it, he would be less "yellow."

What you should be is not yellow at all. If you're supposed to sock somebody in the jaw, and you sort of feel like doing it, you should do it. I'm just no good at it, though...The more I thought about my gloves and my yellowness, the more depressed I got, and I decided, while I was walking and all, to stop off and have a drink somewhere.

Later in the chapter, Holden has an opportunity to sleep with a prostitute, but he can't bring himself to do the deed. Privately, he's embarrassed that he's still a virgin and thinks that, if he could be a little more assertive, he would be more successful with women. However, he argues that, since something always happens when he's on the verge of losing his virginity, it's hard not to get depressed about the whole affair. Even when the opportunity presents itself to him, he still feels awkward and unsophisticated. In this chapter, he lies to the prostitute so that he won't have to lay bare his sexual inexperience before her. He tells her that he's just had an operation on his "clavichord." A clavichord, of course, is a keyboard instrument, popular in the 15th through the 19th centuries.

The trouble was, I just didn't want to do it. I felt more depressed than sexy, if you want to know the truth. She was depressing. Her green dress hanging in the closet and all. And besides, I don't think I could ever do it with somebody that sits in a stupid movie all day long. I really don't think I could.

Notice that he makes an excuse to rationalize his inaction to himself. Basically, Holden is depressed because of his low self-esteem, due to his inability to perform what he believes are the acts of a man.
In Chapter 14, he tells us that he talks out loud to himself when he feels depressed. For example, when he remembers Allie, he remembers his part in excluding Allie from a BB game he and Bobby had planned to have. Since Allie's death, Holden has never been the same. When he gets very depressed, he remembers how Allie never made a fuss about being excluded from the BB game. In order to mitigate his feelings of sadness, he then tries imagining that he did give his brother permission, and this comforts him somewhat.

So once in a while, now, when I get very depressed, I keep saying to him, "Okay. Go home and get your bike and meet me in front of Bobby's house. Hurry up." It wasn't that I didn't use to take him with me when I went somewhere. I did. But that one day, I didn't. He didn't get sore about it--he never got sore about anything-- but I keep thinking about it anyway, when I get very depressed.