A “square from Delaware” was 1930s-1950s slang for someone who was very conventional (and not “hep”). “Let him be as popular as the gigantic. colossal sporty, hip square from Delaware, E. (Nokie) Noko” was printed in the Philadelphia (PA) Tribune on November 11, 1937. “(You’re a) Square from Delaware” (1940) was a Fats Waller (1904-1943) song.
 
The state name of “Delaware” was used simply because of the rhyme. The term “square from Delaware” is of historical interest today.
   
 
11 November 1937, Philadelphia (PA) Tribune, “Around the Schools: Gratz’s Bulldog Bites” by Prof. Seesitall, pg. 8, col. 4:
Let him be as popular as the gigantic. colossal sporty, hip square from Delaware, E. (Nokie) Noko.
   
27 May 1939, New York (NY) Amsterdam News, “Backdoor Stuff” by Dan Burley, pg. 20, col. 1:
“I’m a Square for Delaware, a Lane from Spokane, a killer from Manila and a Home from Rome.”
   
2 March 1940,  New York (NY) Amsterdam News, “Lenox Avenue Is Mecca of Town’s Jitterbugs, by Carl Dunbar Lawrence, pg. 21, col. 3:
So don’t be the “square from Delaware,” just lace up your boots and “dig this jive.”
   
OCLC WorldCat record
“Send Me” Jackson : fox trot
Author: Fats Waller; Rhythm (Musical group)
Publisher: Camden, N.J. : Bluebird, [1940]
Edition/Format:   Music : 78 rpm : English
 
Shazam
Square From Delaware
Fats Waller
(Song lyrics include the following.—ed.)
“Follow my tip
You better get hip
You’re nothin’ but a square from Delaware.”
 
7 June 1940, Springfield (IL) Daily Republican, “Recordings of the Week,” pg. 3, col. 4:
“Fats” Waller
Always high entertainment, “Fats” Waller and his orchestra record in their typical style “Send Me Jackson” and “Square from Delaware.”
     
OCLC WorldCat record
Square from Delaware.
Author: Leonard Feather
Publisher: ©1940.
Edition/Format:   Musical score : English
 
10 June 1943, Chicago (IL) Sun, “What’s a Zoot-Suiter?,” pg. 11, col. 7:
One who dressed normally is labeled a “square from Delaware” who isn’t “hep.”
   
28 August 1943, Miami (FL) Herald, Service Parade sec., pg. 8, col. 2 illustration caption:
SQUARE FROM DELAWARE
...unsoldierly soldier
     
OCLC WorldCat record
Don’t Be a “Square from Delaware”
Edition/Format: Article Article : English
Publication: The Clearing House, v18 n5 (19440101): 273
 
15 December 1946, The Sunday Star-Ledger (Newark, NJ),  “Tricks for Teens” by Nancy Pepper, pg. 37, col. 4:
Drip from Mississip…he replaces the Square from Delaware.
 
24 May 1947, Detroit (MI) Tribune, “Jivin’ with Jennings” by Dick Jennings, pg. 11, col. 4:
Or are you simply a lane from Spokane, or a square from Delaware, acting like a bear on the lam in Times Square.
   
24 November 1970, Record American (Boston, MA), “Aries Usually Proves Creative” by Bert Bacharach, pg. 32, col. 3:
Faded Phrases: “He’s a square from Delaware,” ...
 
University of Delaware Messenger
Volume 10, Number 3, 2001
Square from Delaware a radio classic
He’s one of New York City’s foremost culture vultures, but his preferred nickname remains “The Square from Delaware.”
 
Gregg Whiteside, AS ‘70, chief announcer and manager of radio station WQXR-FM, New York’s premier classical music radio station, sounds European as he rattles off the melodic names of the pieces he plays on the air. In between songs, while verbally sparring with newscaster Sam Hall, Whiteside’s alter ego–the True Blue football fan and UD supporter–never misses a chance to put in a good word for his alma mater.
 
YouTube
Fresh Prince of Bal Air - 72 hours esp.
Miss Dominique
Published on Jul 9, 2013
0:05
“If you’re a square from Delaware and ya got on green underwear then ya better say ‘Oh yeah!’”
 
Google Books
The Routledge Dictionary of Modern American Slang and Unconventional English
Edited by Tom Dalzell
New York, NY: Routledge
2018
Pg. 2013:
square from Delaware noun
an exceptionally naive, conventional person US, 1938
Delaware exists for the rhyme; it is no more or less square than any other state. In the 1930s and 40s, there was a cottage industry in inventing terms along this line of construction—a “clown from Allentown,” a “pester from Chester” and so on. The “square from Delaware” was one of the few that truly worked itself into speech.