The god Mimir was the advisor to Odin in Norse legend. 
His name meant "The Rememberer, The Wise One."
His talents were in logic, wisdom, and counsel.

In the modern age, the gods have urges to appear when they're
most needed, or when what they represent is notably absent. 
Whether they appear or not is entirely up to them, but in the case of FOIL and its judges, the overall lack of rationality was too much for Mimir's natural inclinations.

He took the form of something irrational to inspire rational thought among the judges. 

Once they could see the obvious irrationality
in their life, they would also see that he was a talking dog,
and his work would be done.

As a counseling force, Mimir rarely takes direct action. Instead,
he attempts to provoke thoughtfulness in the people around him,
hoping that logic will provide direction for the correct course. 
However, the judges were far beyond logic. 

Their obsessions had taken over any hope they had of 
being reasoned with, and Mimir—who had appeared suddenly 
and taken the role of their butler and pet—would sound to them like nothing more than the tiny barking dog he appeared to be.

Mimir realized that action had to be taken to get them to see reason,
and understood that only the destruction of their coveted collection
would open their eyes. But he still could not take action directly. (Modern gods are forces of nature and are as incapable of defying their habits as the wind is incapable of not blowing.) 
Instead, as new members arrived to Château de l'Hiver, he attempted to guide their thoughts with musings and stories developed from paintings and objects around the château. Using the furnishings to tell his tales meant that members would be subconsciously reminded of his morals as they walked around. The more logical a member was, the easier it was for Mimir to communicate this information to them.

The first member to fully notice that Mimir was a talking animal
was Kant Merevisi. His reliance on thought let him see right through
the irrationality of Mimir's facade. However, he was not fully
ready to listen to Mimir, instead choosing to pore over his own
obsession: the moon. This blocked Mimir from making any progress
with Kant. The Moongazer suffered from the same type of fanaticism
that the judges did, and none of them could be reasoned with.

Mimir made notes to keep track of members and their mentalities.
He organized his thoughts and approaches, preparing himself for
the best candidates to learn what he had to teach.

Mimir's second breakthrough was with Anastasia Kessler. She was not
perfect, but her intense personal goal to not show weakness—to be 
completely independent—made it easier for her to learn the lessons of Mimir's stories.

But that did not last long. As Ana became more independent, she showed less interest in her Prized Possession, which provoked Grams to steal it. The judges knew that the best way for a person to obsess over their object was to have it taken from them suddenly.

Grams' aged hands could not handle the fragile figurines that comprised Ana's Prized Possession, and it fell and shattered. The impact on Ana was so severe that all of Mimir's progress was undone. Ana left, and suffered a terrible accident on her way home.

A week after Ana's departure, Mimir discovered the most receptive member yet. Florentine Blanc was a self-proclaimed traveler who touted the mantra "less is more" whenever she could. Florentine was logical enough to hear and understand all of Mimir's stories, but was not ready to see him as a talking animal. Her concept of "less is more" was only a light personal guideline, and even she took a small interest in collecting, albeit not to the point of obsession.

Over the course of seven days, Florentine both added to her personal collection and mulled over the concepts Mimir was trying to teach her.

Through Mimir's subtle provocations and her innate curiosity, 
Florentine discovered what the judges were doing in secret. She took
Mimir's final advice to heart and burned the collection the judges had
worked so hard to build.

However, the incident was too traumatic for Florentine. She had seen
herself as a simple traveler with no reliance on physical possessions.
But the week-long adventure, which had almost ended in her death,
made her fear obsessive and reliant tendencies. She feared becoming like the judges. Her once-balanced mentality teetered too far in the direction of "less is more," and she cast the only object that she found truly important—her lunch box—into the fire.

To protect herself from a fate she thought was inevitable—that of
being a possessive judge—she overcompensated and allowed herself to be consumed by her simple mantra. As she left the château, she didn't even bother to bring her necessities.

Mimir, having finally taught the lessons he aimed to, resigned to
barking. He had nothing more to say. And Florentine, in her last moments by the fireplace, saw him for what he appeared to be: a strange talking dog. As she left, he returned to his own realm of existence.

The intense threat—as he sees it—of unreasonable thought was averted, but at the expense of many lives and the psyche of Florentine, the only student who listened to him during his three years at the château.

Inside him is the urge to teach her more—to correct his mistake of
cementing her mind. And though her single life isn't important enough to demand his presence, he still guides her on occasion by leaving little yellow sticky notes.

CLUES:
-Squeaker's avatar is of only his head. Mimir was beheaded during a war.

-Squeaker's collar has a small well on it. Mimir's head resided in a well.

-Squeaker's bed basket is also reminiscent of a well.

-Squeaker mentions wisdom on more than one occasion.

-Squeaker talks about the hidden reasons each member collects, something they probably don't know themselves.

-Roland mentions that Squeaker told him a story about a man in a well (a reference to Mimir's head in the well) and a giant tree (a reference to Yggdrasil, the Norse tree of life).

-Odon talks about what modern gods would look like and says that a
God of Logic or Wisdom would likely appear as something irrational
(like a talking chihuahua).

-Odon also mentions that modern gods would likely appear where they feel they're needed most. In Mimir's case, this would be in a situation where people are incredibly unreasonable.

-It is implied on several occasions that Squeaker is much older.

-The description on Squeaker's Mask says that Squeaker may know some "outdated languages," which subtly implies Old Norse.

-Squeaker's text email is "RimimDog," which is "God Mimir" backwards.

-Squeaker's head spins hypnotically when he talks, an impression that what he's saying is seductively boring into Florentine's mind.

-When telling stories, everything disappears except for this hypnotic
movement of Squeaker's head.

-The Squeaker's Mask is rumored to spin when no one is looking. It is meant to imply that Squeaker is still working his magic on Florentine even when he's not around.

-When gossiping about Squeaker to Kant, Kant complains that he's the only one who is confused about Squeaker being a talking dog. This goes back to him being the most logical person in the château. Florentine comments afterward that he's overreacting, suggesting that she kind of recognizes this to be strange but doesn't see it as a big deal yet.

-There is no scene with both a judge and Squeaker where Squeaker says anything but "arp." This is because the judges are deaf to his reasoning.

-The object left in Ana's safe is a Norse snow globe. Florentine wonders briefly if it belongs to Squeaker.

-The song on the snow globe translates to "The Man in the Well" from Old Norse.

-The song the snow globe plays is also the song that plays when Squeaker tells a story.

-In the final scene of the game, Florentine remarks that she doesn't
know how Squeaker gets into her head, but that he always seems to manipulate her mind.

-Florentine often shows displeasure when references are made to Squeaker being a dog. Even though she knows he's a dog, she sees him as a person.

-After burning her lunch box, she finally calls Squeaker a dog. This is
because she's finally wise enough to see what's been staring her in the face the entire time—an irrational situation.