Yes, my mother remembered the Red Cross giving yarn to the girls in her high school in the 1940s to knit for some WWII related project. I'm not sure if it was squares. She said the girls took the yarn and didn't knit. Well, maybe the mothers were too busy working building ships or planes to teach their daughters to knit.
I believe that most of the WWI and II knitting projects were organized by the Red Cross. I have also heard of something called Knittin for Britain ? Okay UK folks jump in here on that one !
Just fun here are a couple of knitting for the war items. I have also included a hat pattern from WWII that I have used for charity knitting.
http://www.cocoabeachyarn.com/patterns/classicwatchcap.pdf
Beanie No. 212
MATERIALS-- Bear Brand or Bucilla "AA" De Luxe Knitting Worsted, 1 skein, or Bear Brand or Bucilla Standard Brand Knitting Worsted, 1 skein.
2 Pairs (4) Bucilla White 10-inch Double-pointed Knitting Needles, Size 4, Article 3843.
Cast on 96 sts, 32 on each of 3 needles. Join and work in ribbing of k 2, p 2 for 6 inches (about 42 rounds), then work in ribing of k 6, p 2, for 22 rounds. K 1 round plain. Decrease 1 st at both ends of each needle, as follows: * k 1, slip 1, k 1, pass the slipped st over the knitted (slip-k-pass), k to within 3 sts of end of needle, k 2 together, k the last st; repeat from * until 4 sts remain on each needle. Break off yarn, thread end into a darning needle, draw through all sts, twice, tighten and darn in end.