I don't know about divisions within the EU. Whilst the most likely stressor may well be another economic meltdown, I don't know what effect that might have. I think the likliehood of any individual country being kicked out is negligible - a quick check online would suggest that there is no mechanism in the Treaty on European Union for any Member State to be expelled from the EU so once you're in you're in unless you choose to leave - when the Eurozone Crisis kicked off a few years ago the members who were worst affected (predominantly in Southern Europe) were bailed out, so there is no incentive for such States to voluntarily leave (the worst sanction a Member State can face is to be suspended temporarily).
As for any of the major States choosing to leave, in the UK it's a common belief that France and Germany are at the heart of the EU and as such would be highly, highly unlikely to ever want to leave it, quite the opposite in fact, they would wish to see the EU States become more closely integrated until the EU eventually becomes a United States of Europe, with common laws, currency, a European Army, etc, etc...I said in an earlier post I didn't want to get into politics, but it goes with this territory...I'm not advocating for or against any particular position in the Real World here, however I do think it unlikely that any of the Northwestern European States would voluntarily withdraw from something that has taken decades to create. It's a belief in the UK that withdrawal from the EU (also known here as Euro Scepticism) is unique to the UK but I would be interested to hear what other European members think in relation to a possible EU breakup in a Twilight 2020's scenario
With regard to other factors promoting division in Europe, we have religious tensions / terrorism, social unrest (it's only a couple of years since the London riots), and reports of extremist / hate parties gaining ground in different parts of Europe, so there's probably a fair bit to work with, much of it quite distasteful.
Also, I did think that in this scenario the Russians would need to set up some sort of counter to the EU and then of course I remembered that such a thing already existed, the Eurasian Economic Community, or EurAsEC, which in our time line currently consists of Russia, Kazakhstan, and Belarus
http://en.wikipedia.org/wiki/Eurasia..._customs_union