Well, I think that this is not so simple. Being member of the EU, doesn’t mean that you are not independent.
There are certain "requirements" that a country has to meet in order to be considered. There are certain "requirements" in order to maintain your membership. And there are certain criteria that would allow the EU to toss you out as well. So, you tell me how independent that is. You just can't join, you have to assimilate first.
Also, the EU then reserves the right to even influence your domestic policy such as gays in the military. (There are already cases on this subject) The way your country does its banking, trade, and so on.
The EU is not just some club it's an economic answer to the United States. It's a race for global influence and resources.