The term squaw winter denotes a short period of wintry weather occurring in autumn in the northern United States and in Canada.

(However, cf. quotation 3, below, in which the term is applied to a period of wintry weather occurring in spring.)

The term squaw winter was coined after Indian summer, from the fact that, because a squaw winter often precedes an Indian summer, they were seen as constituting a couple.
—Cf. origin of ‘Indian summer’ and French ‘l’été sauvage’.

These are, in chronological order, the earliest occurrences of the term squaw winter that I have found:

1-: From the Cleveland Herald and Gazette (Cleveland, Ohio) of Friday 3^rd November 1837:

Correspondence of the Herald and Gazette.

Columbus, Oct. 30.
We are suffering here under the bitings of what is called a “Squaw winter,” the precursor and companion of the “Indian summer.” The whole valley of the Scioto was mantled with snow on the night of the 25^th inst., and it has scarcely departed yet. The night of the 26^th and the day of the 27^th, were times that will be remembered in this section on account of the unseasonable severity of the weather.

2-: From Zion’s Herald and Wesleyan Journal (Boston, Massachusetts) of 2^nd June 1847—as quoted in the Oxford English Dictionary (3^rd edition, 2021):

It was customary […] for the settlers, at the commencement of the squaw winter, to turn out in companies of thirty or forty men, and hunt bears.

3-: From The Western Whig (Bloomington, Illinois) of Saturday 13^th April 1850—the term squaw winter is here applied to a period of wintry weather occurring in spring:

Quite a squally squaw winter is squeaking around us at this time.

4-: From Things About Home, published in the Monongalia Mirror (Morgantown, West Virginia) of Saturday 29^th October 1853:

We had “squaw winter” on Monday last. There were at least 3 inches of snow!

5-: From the Daily Commercial Register (Sandusky, Ohio) of Saturday 27^th October 1855:

Snow fell in Wakeman township, Huron county, last Wednesday, to the depth of six inches, and in Townsend township to the depth of eight inches—so we are reliably informed. This may be considered a good strong “Squaw Winter.”

6-: From the Wisconsin Patriot (Madison, Wisconsin) of Saturday 3^rd November 1855:

A Snow Storm.

On Sunday morning last, we awoke to find tho [sic] ground covered with snow, and thick white flakes still falling. It was a stormy Sabbath. Nearly three inches of snow fell—but it melted away very quickly. The storm extended all over the State, and has been succeeded by a week of December weather, or “Squaw Winter.” Indian summer has either been drowned out or driven in some muskrat hole.
Well, this has been a singular season—a drouth in May, a frost in August, and a flood in October. “Drive on!”
Another Storm.—We awoke yesterday morning and found it snowing right merrily, with a heavy gale outside our valley.—It held up soon after 9 o’clock. Temperature much milder: flattering prospects of a shower and a thaw. Verily, it has been as cold as that charity which gives its golden eagles to christianize the Heathen in Hindostan, but rarely a dime to flannelize the poor at their very door.—Appleton Crescent.

7-: From The Princeton Clarion (Princeton, Indiana) of Saturday 7^th November 1857:

Snow.—The late cold weather produced both ice and snow. In various part of Massachusetts, New York, Ohio and Indiana, snow fell freely; in some localities to the depth of six inches. The Cumberland Mountains are now mantled in white. That we may call “squaw winter,” after which comes “Indian summer.”

8-: From the Cleveland Morning Leader (Cleveland, Ohio) of Saturday 9^th October 1858:

Squaw Winter.—The “oldest inhabitant” was considerably astonished yesterday morning to wake up and find middle aged autumnal nature preserved in ice. The very “shrewd and nipping air” that was abroad during the night, had blown up a storm and sleet of a very disagreeable description, and Squaw winter rushed in before Indian Summer, which proved itself to be, this time, what our German friends style it, “Old Wives Summer.” The beams of the sun soon melted the ice, and slightly warmed the air, but the searching wind continued all day and made shawls and overcoats a necessary auxiliary to comfort.
Stern winter will soon be upon us and it behooves us to make provision for its advent, with supplies of fuel and winter garments. And let not the poor be forgotten. The coming of winter brings a dismal prospect to those who are destitute of both money and work, and “great will be the reward” of those who give from their abundance to the poor and needy.