"You're like a millennial Natalie Portman. Like I'm V for Vendetta." (1.17). After Gus says this Hazel goes over and watches V for Vendetta with him. Him telling her this starts off their whole relationship. If he never said anything to her though, they probably would not have ever met or dated.
"Hazel and I are going to watch V for Vendetta so she can see her filmic doppelganger, mid-two thousands Natalie Portman" (2.29). Augustus and Hazel watch V for Vendetta on their first date after support group. Augustus introduces Hazel to his parents and shows her his room after a dinner of enchiladas.
This pin is important to Augustus because this is the first movie that him and Hazel watched together. After he met her for the first time, he invited her over to watch this movie because she had never seen it.