Surprise… Fiat currency and central banks in the digital world. Run on banks not likely to be all that effective.
There would be a shortage of cash for a period of time, but to do a real modern bank run you'd need to hit the bond market and cause a liquidity crisis for which France would have to go to the EU to solve.
People looking to financially crash the EU need to take advantage of the structure of the EU. It is kind of like the U.S… except each state can authorize printing of its own U.S. Dollars. This means that while "the euro" exists, so, too, do horribly over-leveraged states (like California) that are financially insolvent and effectively riding on the credit of other nations.
France is one of those. Greece is/was one (that was fun)…