I think the basic problem is one of misunderstanding. For quite a while, mainly due to the Press, people have talked about EU citizenship, there is no such thing as EU citizenship, there is EU membership voluntarily, but not EU citizenship (citizenship is involuntary you're born into it).
The second misunderstanding is regarding three entities which are treated as the same but are, in fact, completely different - country, nation and state. Scotland is a country, that is to say that there is a geographic landmass which is recognized both internally and externally as Scotland. Scotland is also a nation, that is to say that certain institutions exist which are uniquely Scottish in the country known as Scotland. Scotland, however, is not a state. So what's a state.
The definition of a state as recognised throughout the world, except curiously in the United States is this - a state is a self-perpetuating entity which is the sole repository of power and prerogative of rule within its own sovereign territory and it maintains that power by the exclusive exercise of legitimate violence. In other words, a state is all of a government's institutions which include military force. Clearly Scotland does not fall into that category.
Now consider Mr. Barroso's words. Scotland would be a new state and would have to re-apply. In short, the Referendum is not about creating a new country or a new nation, but a new state.