I was going to add this as a comment but it turned out to be too large.
OED(2) has an entry for squeaky (adjective) and definition c talks about squeaky clean.
Here they have a citation of use from 1975, which is the figurative sense of squeaky clean meaning beyond reproach. That seems very odd to me, I'm sure the Fairy adverts were before 75 and the figurative squeaky clean is based on the physical squeaky clean.
So after a little poking about, the earliest reference to squeaky clean I can find is from an advert in volume 51-52 of the 1936 publication of Motion Picture; a magazine about movies.
Presumably the magazine is still under some copyright so here is a link. The cover page is so badly scanned that it's not possible (for me) to see if the dating is correct but volume numbers put it about the mid-30's. As you can see it is an advert for hair products from a company called 'Duart'. It seems they still sell hair products and this trademark dates it around the same time as the publication.
So it looks like squeaky clean did not originate with dirty dishes, but with clean hair.