“Square meal” is a term that appears to have originated in California by 1856. It is sometimes claimed that “square meal” comes from England’s Royal Navy (where meals were allegedly served on square wooden plates), but the many 1850s and 1860s citations indicate a California origin.
 
A “square meal” is a proper, substantial meal—an extension of one definition of the word “square.”
   
   
The Phrase Finder
A square meal
 
Meaning
A substantial, nourishing meal.

Origin
It is widely reported, by tour guides and the like, that this originated from the Royal Navy practise of serving meals on square wooden plates. Such plates did exist and so that sounds like a plausible story. There’s no evidence to support it though.
(...)
The word square has many meanings, including ‘proper, honest, straightforward’, and that’s the meaning here. This isn’t a rectilinear meal on right-angled crockery, but a good and satisfying meal.
 
The phrase is of US origin. All the early citations are from America, including this, which is the earliest print reference I have found - an advertisement for the Hope and Neptune restaurant, in the California newspaper The Mountain Democrat, November 1856:
 
“We can promise all who patronize us that they can always get a hearty welcome and ‘square meal’ at the ‘Hope and Neptune. Oyster, chicken and game suppers prepared at short notice.”
 
(Oxford English Dictionary)
square, n.
Elliptical uses of the adjective: a square meal (orig. and chiefly U.S.).
1882 O. MERDIAN Let. 20 Sept. in Frontier (1930) X. 252/1, I went in..and had some dinner..ate a square & talked awhile.
1927 ‘J. BARBICAN’ Confessions Rum-runner xxiii. 260 We sure was hungry for the dough, for it was weeks since we had roped in our three squares a day.
1962 ‘E. MCBAIN’ Like Love ii. 21 But he had had a clean bed to sleep in, and three squares a day, as the saying goes.
1979 ‘H. HOWARD’ Sealed Envelope x. 135 Mine was a lousy job. There must be a better way of making three squares a day.
   
7 February 1856, Georgetown (CA) News, pg. 3, col. 6:
If you or any of your readers should visit our town, call at the United States Hotel, for every person goes there to get a square meal.
(A letter by Pike from Newton, California.—ed.)
 
15 March 1856, Butte (CA) Record, “Dry Times,” pg. 4, col. 2:
Rendered doubly suspicious by the pressure of the times, the dependents on their wits are often obliged to resort to strategem and stealth for a “square meal,” and as frequently as banked, and obliged to go hungry.
   
Google Books
February 1857, Hutchings’ Illustrated California Magazine, “The World in California,”  pg. 340:
His weekly supply of “grub,” and his one “square meal” on Sunday, he will have;...
     
27 July 1857, Daily Globe (Washington, DC), pg. 1:
A Mariposa paper makes the following interesting announcement. Those “three square meals” ought to attract a large crowd:
CAMP MEETING.—A camp meeting will be held at Cathay’s Rancho, on the 6th of August next. Every person is invited to attend—but they are requested to bring them blankets, as a shed will be built for them to sleep under. The ladies and their families will be accommodated in the surrounding houses. Three square meals per day will be furnished without money and without price.
 
22 August 1857, New York (NY) Daily Times, pg. 4:
If he does, they’ll learn how to enjoy a good square meal when they get back, if they live long enough.
 
Newspapers.com
17 September 1857, The Plumas Argus (Quincy, CA), pg. 1, col. 5:
HOTELS
PLUMAS HOUSE.
ELIZABETHTOWN.
(...)
... try us if you think we are gassing, and see if you don’t get a “square meal” and quiet slumbers.
 
13 November 1858, San Francisco (CA) Bulletin, “Letter from the Frontier, Lower California,” pg. 1:
They look sleek and fat, as though they “fared sumptiously every day;” when, in fact, they don’t know what a good, “square meal” is.
   
13 January 1859, San Francisco (CA) Bulletin, “Notes of Travel on the Frontier of Lower California,” pg. 1:
My wife knowing there was no grub between our house and La Grulla, give them some bread and wine to stay their stomachs till she could provide them a substantial dinner, which they said was the first square meal for a long time.
 
Travel and Adventure in the Territory of Alaska
By Frederick Whymper
New York, NY: Harper Brothers
1869
Pg. 310:
A good substantial repast is known as a “square” meal all over this coast, and the term is applied to many other things.  A “square” drink is a “deep, deep draught,” and a good “square fight” is an encounter or “muss” where the opponents were in earnest.  Some of these terms are common to the “Western” States and outlying “territories,” but can not be regarded as full-blooded Americanisms.  They attract just as much notice from “Eastern” men travelling in California as they do from Europeans.
 
10 December 1874, The Cultivator & Country Gentleman, pg. 799, col. 1:
SLANG.—We allow ourselves to say of a rich man that he has got “stamps;” of the drunken man that he is “tight,” or “boozy;” of anything that pleases us or is satisfactory that it is “stunning;” “awful” is considered a better word than very, and we are awful cold, or hot, or sick, or jolly, as the case may be; it is finer to say “you bet,” than to answer a simple yes; everything that annoys us is “infernal” or “beastly;” bank bills are “greenbacks.”  I heard a lady in a good society say recently that her dressmaker had disappointed her, and that in consequence she was “regularly up a tree;” we threaten not to humiliate or mortify a man, but to “take the starch out of him:” we rack our brain to invent slang words for various drinks, and bring out such names as “forty-rod,” “tangle-foot,” “rot-gut,” “blue ruin” and “Jersey lightning,” words that would puzzle a foreigner; a man is not cheated, but “done brown,” or “bamboozled;” railroad conductors do not steal (in fact we are getting a little sensitive about using the word), but “knock down;” bank cashiers do not swindle and steal, but commit “irregularities;” we hear of a house being “burgled,” and that two foot-pads “went through” a belated traveler; a fair dealer is spoken of as “a square man,” a most wonderful lusus naturae; a substantial dinner is spoken of as a “square meal;” we hear invitations given, not to take a drink, but to “hoist in some poison;” anything antiquated or exhausted (Col. 2—ed.) is “played out;” an insufficient excuse is said to be “too thin,” or we are told that it “will not wash;” we buy stocks on a “margin,” or sell them “short,” or “bull” the market; or “take a flyer,” or “scoop in a long line of stocks;” we do not stake a sum of money, but “bet our pile;” after a convivial party we next morning find ourselves “precious seedy;” our railroad trains “telescope,” or a “Pullman” breaks a wheel; a party of rowdies “clean out” a drinking-saloon; a big man threatens to “wipe out” a little one; we do not outwit or circumvent another, but “euchre” him; we “take the shine out of” a rival, and “fix his flint” for him; a carpenter “runs up” a cheap house in a week; an investigating committee in Congress “whitewashes” the character of some defaulter, and so on and so forth in all the departments of business and trade and social intercourse we permit ourselves to use words and phrases which are of no authority, often vulgar and always needless.—American Homes.
   
OCLC WorldCat Record
Square meals : a cookbook
Author: Jane Stern; Michael Stern
Publisher: New York : Knopf : Distributed by Random House, 1984.
Edition/Format: Book : English : 1st ed