Of American-English origin, the adjective squeaky clean (also, with hyphen, squeaky-clean) means:
– originally: (of hair) washed and rinsed so clean that it squeaks;
– by extension: completely clean;
– figuratively: above criticism, beyond reproach.

The earliest occurrences of the adjective squeaky(-)clean that I have found are as follows, in chronological order:

1-: From Keeping Hair Beautiful, by Judith Merrill, published in the San Francisco Examiner (San Francisco, California) of Tuesday 30^th August 1938:

No amount of shampoo can leave your hair clean if you don’t rinse it thoroughly.
SO WITH WARM WATER, rinse three times, or until the rinse water is clear. The warm water leaves your hair lustrous, while cold is apt to dull it.
Three good rinses, then, will leave your locks “squeaky” clean.

2-: From With a Huff and a Puff Your Hair Is Blown Clean, Bright, by Dorothy Randall, published in the Pittsburgh Sun-Telegraph (Pittsburgh, Pennsylvania) of Thursday 3^rd April 1941:

Directly following the air-cleansing, double handfuls of dry wood fiber like sawdust are rubbed through your hair. It is delicious—smelling of camphor, herbs and all sorts of woodsy scents. This method is somewhat like that of cleaning a fur coat—the soil removed without applications of liquids which may mar the natural brightness and sheen.
When the “sawdust” is completely through your hair, the blowing is resumed until every vestige of it is removed.
You find yourself shimmeringly, squeaky-clean (remember how hair “squeaks” when clean?).

3-: From Danger of Scrambled Egg Rinse Eliminated, by Dorothy Randall, published in the Pittsburgh Sun-Telegraph (Pittsburgh, Pennsylvania) of Wednesday 18^th June 1941:

The hair is moistened with lukewarm water and the bar of egg shampoo soap is rubbed over it. The resulting lather is a mass of bubbly egg white which cleanses away every trace of grime, dandruff or oil. Your hair is left Squeaky-clean, beautifully glossy and vibrantly toned up.

4-: From the column Beauty and You, by Patricia Lindsay, published in the Brooklyn Eagle (Brooklyn, New York) of Monday 10^th August 1942:

If you use a good shampoo—one which leaves your hair span, squeaky clean and glossy—I see no necessity for using a lotion to set it.

5-: From A Salon for the Busy […] Workers With No Time to Visit Coiffure Expert Achieve Fair Facsimile at Home, published in The Kansas City Star (Kansas City, Missouri) of Thursday 27^th August 1942:

Shampoo your hair with a liquid preparation which has a hair conditioner added to it […].
Two soapings of this, a vigorous massage with fingers, and three thorough rinsings, make your hair “squeaky” clean and manageable.

The earliest extended uses of squeaky(-)clean that I have found are as follows:

1-: From an advertisement for Toss, published in the Warren Times-Mirror (Warren, Pennsylvania) of Thursday 20^th February 1947:

TOSS TAKES THE DRUDGERY OUT OF DISHWASHING!
This modern cleaner makes your dishes shine and washes glassware “squeaky” clean. Works with LIGHTNING EFFICIENCY!

2-: From an advertisement for Kentucky Utilities Company, published in The Lexington Leader (Lexington, Kentucky) of Sunday 9^th March 1947:

The story of Aladdin and his wonderful lamp has enthralled children for centuries. But the amazing Genie, that all powerful servant of the lamp is no longer just a character in a fairy tale. Wherever electricity is found, Genie with the light brown whiskers has become a household servant.
Instead of rubbing a lamp, our modern housewife flicks a switch! “Genie, wash the dishes.” And Presto! the dishes, cutlery, and pots and pans are washed, rinsed and dried squeaky clean.

The earliest figurative uses of the adjective squeaky(-)clean that I have found are as follows:

1-: From O’Neill Finds Cause for Cheer Despite Loss in Opener, the account of a baseball match, by Leo Macdonell, published in the Detroit Evening Times (Detroit, Michigan) of Thursday 22^nd April 1943—the comma between the adjectives squeaky and clean is most probably a typographical error, as I have found no other occurrences of squeaky + comma + clean:

CLEVELAND, April 22—Steve O’Neill need not feel downcast, and probably doesn’t, over the defeat of his Tigers at the hands of the Indians in the first game of the season.
It was a squeaky, clean victory Lou Boudreau’s men captured—a game decided by a fly ball.

2-: From 1951’s Pioneer Day Election Promoting Is Said Best Ever, published in The Chico Enterprise-Record (Chico, California) of Wednesday 9^th May 1951:

Tuesday morning a two-column tabloid urged the election of a sheriff candidate who allegedly cleaned up a Nevada community until it was “squeaky clean.”

3-: From the review of The Merry Widow (1952), a U.S. film starring the U.S. actress Lana Turner (1921-1995) and the Argentinian-born U.S. actor Fernando Lamas (1915-1982)—review by Patrick Skene Catling, published in The Sun (Baltimore, Maryland) of Saturday 6^th September 1952:

Lamas kisses Turner some two dozen times during the proceedings; and Turner kisses Lamas right back—once with such enthusiasm that her upper lip, when finally retracted, is actually moistened, a touch of realism rarely achieved before the cameras of the Culver City film factory ¹.
However, The Code ² prevails, you may be sure. The general effect is squeaky-clean.

¹ The Merry Widow was made at the Metro-Goldwyn-Mayer film studios, located at Culver City, in Los Angeles County, California.
² This refers to the Motion Picture Production Code, i.e., a set of moral guidelines applied to U.S. motion pictures. It is known as the Hays Code, after William Harrison Hays (1879-1954), who, from 1922 to 1945, was the Chairman of the Motion Picture Producers and Distributors of America (MPPDA).