‘Lynneguist’ (Lynne Murphy), in her always-entertaining blog separated by a common language, currently discusses British and American words such as squidgy podgy pudgy splodgy dodgy.
It turns out that squidgy, which is apparently British but not American, dates back only to 1973 in its current meaning of ‘moist and pliant; squashy, soggy. Esp. of food’. In a different meaning, ‘short and plump; podgy’, it has been around a little longer, since 1891, and is first attributed to Kipling. (Why did Princess Diana’s lover give her the nickname Squidgy? She was not short and plump, but rather tall. ‘Moist and pliant' would seem to fit the bill better.)
Kipling also supplies the first OED citation of squdgy [sic], which Lynne aptly characterizes as “a word that looks like a typo”.
Why does it look like a spelling mistake? Because in English the letters <qu> are a digraph representing kw or sometimes k, and are always followed by a further vowel letter: quantity, question, quick, quorum, obloquy. The vowel can be silent, as in antique, but it’s got to be there. But in squdgy it isn’t.
Excursus: the third word in the name of the village Stow cum Quy in Cambridgeshire looks odd, too, but actually fits in with the usual pattern. (Most sources say Quy is pronounced kwaɪ, though Wikipedia currently disagrees and goes for kaɪ.)
Since squdgy is pronounced (by those who use it, of whom I am not one) as ˈskwʌdʒi, logically it ought to be spelt squudgy. But the doubled u looks odd (despite the familiar Latin words equus and vacuum) and is hence avoided.
There seem to be no other words that contain the sequence of sounds kwʌ.
The historical explanation is no doubt to be found in the fact that ʌ used to be ʊ. The sounds kw can be followed by most vowel sounds (queen, quick, quest, quack, qualm, quantity, quorum, quirk, equate, quote, quite, (quoit), queer, square), but never by ʊ uː aʊ ʊə. One of these, aʊ, may be an accidental gap, but the other three seem to show that you can’t have a close back vowel after kw.
The only other word spelt with qu followed immediately by a consonant is Qu’ran, Quran, alternative spellings for Koran. Here of course the letter q constitutes a transliteration of Arabic ق, as in the u-less Arabic borrowings qat, Qatar, qibla.
My favourite Chinese beer, Tsingtao / Qingdao 青岛 1tɕʰɪŋ 3tɑu, English ˌtʃɪŋˈdaʊ, has nothing to do with the case.