What are some examples of Holden acting insecure or becoming paranoid that people are judging him in "The Catcher in the Rye"?
There book is full of examples of this. Holden tends to judge other people so they can't get close to him and judge him or hurt him. When he visit Old Spencer in the beginning of the book, rather than listening to what Spencer has to say, he just judges Spencer because Holden doesn't want to hear that he's doing anything wrong.
Throughout the book, Holden participates in activities that would make him seem older: drinking, smoking, living on his own, hiring a prostitute (even though he admits that he's confused about sex). Holden wants to seem more like an adult--in a superficial way--so that others around him don't judge him as being "too young". He's insecure about where he fits in the world, so he tries to act older, when he's really less mature than he needs to be.
When Holden is on his date with Sally, he becomes insecure when she meets up with George at intermission. Holden may be truly distasteful of phony conversations and meetings, but Sally and George represent people who grew up in the "right way", as phony as they may be. At the end of the date, when Sally turns down Holden's plan to run away, he begins to insult her in order to save his feelings. He opened up to her in a way that he hasn't opened to anyone else, and she completely shoots him down. In order not to feel hurt, he needs to put her down to show that her opinions weren't worth anything in the first place.