Squirrels are pretty hairy. No need to make coats for them.
There is a market, but it involves someone with a rather cruel reputation whose name is similar to a mythological monster.
China uses dog fur for clothing so there is a market for dog fur. You can read many incidents about a retailer pulling their coats from sale in the USA after it becomes public knowledge that they have some coats with dog fur on them.
Dog fur has a reputation of having strong hair that doesn’t shed (in the finished product, as opposed to a living animal). Good quality stuff.
Why do you think people have squirrel rifles? To harvest food. I always stuck to rabbit, pheasant, and deer, but I know plenty of people that hunted and ate squirrel. I once worked with a guy that talked about the importance of shooting them between the eyes to avoid ruining any meat. I think he was from Kentucky.
Even the coyote is marketed as prairie wolf.
I think most euthanized dogs and cats are incinerated or go to the rendering plant.
Of the people I have known that trapped, they mostly caught muskrats. I seldom see muskrat coats advertised. Another name?
They used to be sold (dyed) as “Hudson seal”.