FWIW my opposition to the EU has everything to do with wanting to live in a state that's governed in a democratically accountable way, and very little to do with nationalism and the usual ugly 'Little Englander' stereotypes that get thrown at eurosceptics.
The big-picture reasoning behind the EU, for its supporters, is roughly that 1) the nation state should be encouraged to wither away as nations just encourage ugly xenophobia and besides the nation state is obsolete in an age of globalisation, and 2) we need larger trading blocs to cope with the new economic realities of the modern age.
To 1) I say: that's a long and complex debate, but the biggest objection I have to the idea of abandoning nation states is that if we're not careful we'll end up abandoning democracy as well. If it is indeed true that the age of the nation state is over, we should be explicitly debating what should replace it, and not sleepwalking into a situation where that's being decided for us by a bunch of Commissioners we can't even elect let alone get rid of.
To 2) I say: the EU's track record so far doesn't fill me with confidence. Its shocking handling of the eurozone crisis (50% youth unemployment in Spain, anyone?) and the fact that pretty much every region in the world is now growing again, except the EU suggests that even if it were true that we need larger trading blocs, the EU is doing a pretty poor job of being that bloc.
Proponents of the EU say that economically we'd suffer outside it. But the EU's record of having a positive effect on members' economies is just not that great. Visit Greece if you're unconvinced. And I'm just not sold on the idea that 28 nations, with often conflicting interests, can act quickly and decisively to address common challenges and crises as they come up. Witness how rapidly the refugee crisis disintegrated from 'European solidarity' to national interests again once the chips were down.
The EU was a great idea, and I'm 100% behind the ideals it represents, or at least that it represented. But its track record of delivery is poor and getting worse. I'm just not convinced that it's worth giving up the sovereignty of our own Parliament to a supranational institution that's shown itself so bad at delivering.