The word squirrel in English was borrowed from French centuries after the Norman incursion in 1066
The original English term before that was acquerne.
Pronounced [ˈɑːkwern] (but who knows maybe people would later have ended up saying [ˈæːkwern]), it’s cognate with the German word Eichhorn (usually in the diminutive: Eichörnchen, which looks like ‘oak croissant’, but isn’t). Both the English and German terms come from the Germanic *aikwernô composed of the roots for ‘oak’ and ‘squirrel’, perhaps differentiating a squirrel that prefers oak trees or being simply pleonastic (*wern ‘squirrel’ by itself may have gotten overlooked too easily or clashed with a more salient homonym at some point in the history of West Germanic). The Indo-European form was *wer ‘squirrel’, but interestingly had a homonym meaning ‘burn’. Given that the species of squirrel common across Eurasia (including the edges of areas of proposed non-Anatolian Indo-European Urheimat) is the red squirrel and looks like its ears are on fire, it’s fun to think there may be a connection there.
It’s possible that acquerne and squirrel competed for a time before the Middle English term was abandoned for the trendier French alternative.
Important to note: acquerne and acorn come from separate, unrelated roots. Acorn is from the Old English ǣcern which referred to the mast (nut fruit, usually accumulated on the ground) of any tree and comes from the Indo-European *ħógeħ ‘berry’.