Well Green Day is strongly opposed to the current administration. But they were also rather opposed to the previous administration. At least that is what I gathered from the one hour special I saw on VH1 at 3 am.
Generally speaking, Green Day belongs to a counter-culture movement. Something they freely acknowledge.
Wether or not that makes them anti-American is a matter of some debate. From what I gather Green Day is stongly supportive of individuality. The idea of individuality is a very American ideal. In fact, American care more about individuality than any other society on Earth.
This is somewhat problematic as societies are built on conformity. In fact, the survival of all people on the Earth is dependent on conformity. All people have strong psychological and biological needs to conform and belong to a group. Just about everything we use on a daily basis is thanks to conformity. Little things like language and all forms of communication.
The conflict arises that Green Day is supportive of ideals that are in a very real sense threatening to many groups in Americans. Thus they can be percieved as anti-American. In Green Day's mind they could be seeing themselves as being very American simple because they belong to a very different group.
This is the kind of stuff that Socio-Cultural Anthropologists, Sociologists, and Social-Psychologists live for. (Don't ask me to explaing the difference between them.) Overall, the modern Western world is experiencing a large amount of fracturization as societies, groups, and cultures get blended about.
Historically, the end result is lots of war and bloodshed.