My mom was part of a girls barbershop quartet in the late 1940s – they sang radio jingles and eventually some TV ads and performed at different venues in around Cincinnati. She used to sing that mosquito song to us when we were kids accompanied by the ukulele. Here are the lyrics that she used to sing:

I’d like to know where mosquitoes go when the wintry breezes blow.
Whatta they wear for their underwear, for their underwear? I don’t know.
He’s a young humdinger, sharpens up his stinger for a warmer clime.
Where do mosquitoes go, I’d like to know, in the winter time?

Zing! Caught another Skeeter! Zing! Big as a kangaroo. zing! Old man eater, bit me right where I can’t tell you.
Couldn’t go in the dining room, couldn’t go in the kitchen, couldn’t go where there’s company cause it’s always itchin.’
Tell your troubles to the man, help you solve the puzzle:
Where do you go in summertime unless they wear a muzzle?

I’d like to know where mosquitoes go when the wintry breezes blow.
Whatta they wear for their underwear, for their underwear? I don’t know.
He’s a young humdinger, sharpens up his stinger for a warmer clime
Where do mosquitoes go, I’d like to know, in the wintertime?