#hap-fräui

Sure thing!

Squirrels apparently mean luck in Norway, but in Sweden, they are a bad omen. If a squirrel crosses the road, you need to spit three times to ward off evil. 

Squirrels also show up in many tydor. If a squirrel looks in through the window, it means that someone is going to die. In some regions, squirrels forebode disease and/or fire. A very large number of squirrels meant war or famine.

But squirrels have also be used for medicinal purposes - different parts of the squirrel would cure dizziness, epilepsy, and eye diseases. 

It is possible that this had something to do with the colour of the animal - red was associated with magic (good and bad). Squirrels also seem to have been associated with fortune telling.

On Gotland, squirrels seem to have been associated with havsfrun:

“Hap-fräui (’sea woman’)* »moves nest» four times a year, and each time, she brings all fish and all her children with her, and they [the children] sail out on the sea on a bark with the tail as a sail: before [this], they [live] on land and are called squirrels.”

Tofta, Gotland, Sweden

In another story, the squirrels are livestock rather than children: 

“A skipper once was sailing on the wild sea, with good wind and full sails. But then the ship suddenly stopped and stood still. The skipper and his crew barely had the time to start wondering about this, when a young, beautiful woman entered the ship’s deck and asked the skipper for some salt. The skipper, who now understood that he had the Haffruen (’the sea woman’) herself on his deck, was at first astonished, but soon found himself and promised her that she would get what she had asked for. But then he asked her what [vessel] he should put the salt in. She replied: » Just throw it overboard, and I’ll take care of it.» — Well, after it had been done according to the wish of Haffruen, she offered to pay the skipper for it; but when he didn’t want to accept any [payment] for such a small [gift], she asked him to come visit her. The skipper anxiously climbed after Haffruen, down the rope ladder, and soon found himself standing on her wide and beautiful stairs made of marble, — And now they were already deep down under the waves [of the sea], in Haffruen’s realm, were everything was nice and luscious, with gardens, courtyards, and other things, and first and foremost, her magnificent crystal castle. She asked him to come in, and [with surprise] he looked at all of the beautiful glass halls, and the rooms that were decorated with gold and pearls. Now Haffruen handed the salt over to her servants, and these used it to cure the meat from the part of her livestock that had been slaughtered. [The livestock] were just squirrels, magpies, and other animals that belong to the devil.  — Then she continued through all of the rooms, until she came to a particularly beautiful chamber that had a pretty little table standing on the floor. When [the skipper] happened to take a closer look at [the table], he saw that a shiny little sewing needle had been inserted into it, and that the needle was threaded with a fine silk thread that seemed to disappear up into the crystal roof. This made him wonder, and he asked Haffruen what it was supposed to be. She replied: » Your ship is tied to this weak silk thread; but since you willingly gave me what I asked for, you won’t be troubled by it, but return to your ship, and then always have good winds and sail with luck!» — And with that, [without the skipper realizing what had happened], he was soon back on his ship, which was sailing with its regular, good speed again. He returned back home safely with his ship and [the ship’s] load, and from then on, he always sailed with prosperity and success.”

Visby, Gotland, Sweden

To me, the last story seems more like a fairy tale than a sägen. Mainly because of its “literary” style. But saga vs. sägen is not black and white - there are what we call “sägenartade sagor” (’fairy tales with the characteristics of a sägen’). 

In Norse mythology, Ratatosk delivers insults from the eagle Hräsvelg to the dragon Nidhögg, and vice versa. 

Regional names for squirrel include: egern, ekorre, ekorn, ikorn, ikorre, ikôrning, iköne, ikane, ikanne, igane, igare, ijärn, ejärn, fnatt, furufnatt, ferefnate, gråskinn, granbjör, granbjörn, granoxe, kurre, kårrä, and tallbjörn. 

Sources:

Vänersborgs museum

SR

Carl-Herman Tillhagen, Folklig Läkekonst