In a book called Old Names - New Growth: Proceedings of the 2nd Aspens Conference (lovely title) there is this entry by Hans Sauer and Ulrike Krischke on quercus, oak, eiche : acweorna 'squirrel', 'eichhörnchen' should probably be placed here, too: according to Pokorny (p. 13F) its etymological root is not *aig- 'oak', 'Eiche' but *aig - 'swinging (violently)', '(sich heftig) bewegen, schwingen.' Still, secondary motivation in OE might have ben ac. Cf. Kluge s.v. 'Eichhörnchen' who connects the word to germ. *aik-'oak'".
Plus, Anatoly Liberman, who is a professor of Humanities at the University of Michigan, writes a word origin blog for the Oxford University Press and has a book out for laymen that is probably worth buying if you are interested in this sort of topic. Word Origins.... And How We Know Them. He wrote that the first German name for the squirrel was eihhurno and the first part of the word, the 'eih', meant 'oak' or was just 'eih'. In other words, it perhaps had two meanings and associating it with 'oak' does not "militate against good sense". The second part of the compound, he wrote was 'hurn' "coinciding with the word for 'horn' by chance." So the modern German word for squirrel would be 'little oak-horn'. He also pointed out that, although it sounds like an odd name for the wee beastie, the Russian name for squirrel is belka, which the first syllable 'bel' meaning white. I have not seen a lot of white squirrels.
If Liberman is correct, then the comment by Cleasby and Vigfusson is questionable. We shall have to wait until I get a chance to see what the entry in the DOE is. Meanwhile, here is a link to Liberman's blog. There are some choice words discussed in the recent past.