It is interesting to me that the German and Old Icelandic names for squirrel are eichhorn and ikorni which seems like the Germanic peoples developed an entirely different name for the species but the Cleasby-Vigfusson Icelandic-English Dictionary states that it is a corruption of the Greek name which all European languages borrowed. They list acvern as the Anglo-Saxon version but 'ac' is 'oak' and eiche in Modern High German is 'oak'. Maybe 'oak' equals 'shade'? Cleasby and Vigfusson speculated that 'ikorni' in the heathen poem Grimnismal 32 was a later addition and spoiled the meter. As well, that Rata-tösker might have not just been the squirrel's name but also the species name for rati (the climber?( sic.)) and tösker for 'tusk' or 'sharp teeth'. Then the squirrel's name was "Climber the squirrel" but this is speculation on their part.
So, if few Normans came over with the Conquest and England remained essentially Anglo-Saxon, why did acvern not survive? It seems like we kept much of the old Anglo-Saxon words but really we kept only a few hundred common nouns. Most words were replaced by the French terms and squirrel was one of them but it is interesting that the Greeks got to name them because I do not imagine forests when I think of Greece nor do I imagine they have squirrels. Goats, lots of goats, but no squirrels.