Strangely both arguments are correct, SueA, although his choice of Deutsche bank as an example was perhaps not the best one to use.
Most of the economic experts I tend to look to for ideas believe that the Euro will end because Germany pulls out of it not one of the weaker southern European countries. This is supported by history. The Soviet Union traded in Roubles. The satellite states bought more from Russia than Russia bought from the satellite states. This meant a flow of Roubles from the satellite states to Russia. To balance the books Russia used something called the "Transfer Rouble" which flowed back to the satellite states from Russia. Satellite states could not actually spend Transfer Roubles but whenever Russia bought anything from a satellite state Transfer Roubles would flow back. This is regarded as an accounting artefact but in reality is a hidden debt by the satellite states to Russia. Eventually Russia decided it could not stand this ever increasing hidden debt any longer and abandoned the Transfer Rouble.
The Soviet experience is exactly mirrored in the Euro. Southern European states buy more from Germany than Germany buys from the southern states. This imbalance is dealt with in the ECB books by Target 2 transfers from Germany to the southern states. They are the Euro equivalent of the Transfer Rouble and are also a hidden debt, dismissed by the ECB as purely an accounting artefact. This graph
shows that the Target 2 transfers from Germany to southern states, especially Spain and Italy, is now approaching €1Trillion, which is a lot of hidden debt. Germany cannot stop the use of Target 2 transfers as they are under control of the ECB but it can abandon them by abandoning the Euro. If the Euro collapses each country will adopt its own currency and Euro in banks in each country will be converted to that country's new currency. When that happens the new Deutsch Mark will appreciate rapidly and the new Escudo, Peseta, Lira and Drachma will rapidly depreciate. People with their money stored in a German bank will see its buying power increase while those with money in local southern banks will see its buying power decrease. That is what Varoufakis was describing and I agree with him.
Completely separate from the flaws in the Euro currency system there is the systemic risk to the world financial system. We are told that the underlying cause of the 2008 financial crisis was that the balance sheets of the major banks were too big, the amount of debt, government, private and commercial was too big and the amount of money committed to derivatives was too big. Since the financial crisis all these parameters have got bigger, in the case of derivatives by an extraordinary amount, but we are told that the financial system is safer. How can that be? Banks have been forced to increase their capital reserves, which is supposed to make them safer, but that increase is a flea bite compared to the worsening of the other parameters. Also what classes as capital reserve is highly dubious. What will trigger the next crisis, when it will happen and how it will develop is completely unknown but the crisis is already there waiting to happen. Why single out Deutsche Bank? For a number of reasons. A possible (probable?) trigger is the collapse of the Italian banking system, which is completely insolvent, like the Greek banking system but much bigger, and Deutsche bank has loaned vast sums of money to Italian banks. Deutsche Bank could not withstand a large-scale default by Italian banks. Another possible (probable?) trigger is the increasing default level in auto-loan debt and corporate bonds. These have all been sliced and diced into derivatives, like sub-prime mortgages were back before 2008, and Deutsche bank has the biggest derivative book in the world I think. Finally Deutsche bank is the most interconnected bank in the world so if it collapses all the major banks will collapse. It is doubtful if the central banks can stop a total collapse this time as they did in 2008 and there will be a massive destruction of wealth.
If you think the first scenario will occur first then holding Euro in a German bank makes sense. If you think the second scenario will occur first then getting out of currency altogether and into tangible assets is the only option, e.g. precious metals, precious stones, fine art, property etc. It has to be something real and tangible outside the banking sector. For preference you should have it at home, i.e. if you don't hold it you don't own it. That has to be balanced against the risk of burglary but don't use bank safety deposit boxes for storage, use vaults outside the banking sector. If you don't have any savings at all or you don't believe either will happen, just carry on living your life and ignore what is happening in the world of finance.