"écureuil" is my mnemonic, but apparently they are not related at all :D
According to Wiktionnary, "écureuil" is from greek σκίουρος (squirrel), that has for etymological meaning something like "shadow-tail", because a squirrel can do shadow with its tail... Yeah... So σκίουρος gave the latin word "sciurus". From this I let you imagine how it gave modern french word "écureuil" and english word "squirrel".
Apparently, Ekorren, on its side, with the german word "Eichhörnchen", comes directly from some proto-indo-european word which had already back then the same meaning it has today.