Summary: In table tennis, going to an odd number is part of removing serve advantage. For badminton and squash, the earliest rationales are probably lost to history, but they both started from a scoring system documented as early as 1872. Otherwise, this seems to be an old tradition not limited to racket sports, though their stability probably helped maintain the tradition to the present day.
After way too much research into this, I've convinced myself that scoring to an odd number is a very old tradition not limited to racket sports. In fact, occurrences of this tradition already appear in Charles Cotton's The Compleat Gamester (1674) which according to Wikipedia is "one of the earliest known English language games compendia." He describes the scoring of Trucks, a billiards variant (pg. 41):
The Game because it is sooner up than Billiards, is Nine, and sometimes Fifteen, or indeed as many, or as few as you please.
And the card game All Fours (pg. 111):
This Game I conceive is called All-fours from Highest, Lowest, Jack, and Game, which is the Set as some play it, but you may make from seven to fifteen, or more if you please, but commonly eleven.
If that isn't enough, I also found too many sources from 19th century England to list. Notably 11, 15, and 21 are incredibly common, 9 a bit less, and other numbers are rare or non-existant. To convince yourself, search Google Books for racquets (also rackets), fives, quoits (also quoiting), bowls and less frequently, shuffleboard (also shovelboard). I did notice though that the older the source, the more it tended to either leave out the game score or say it should be agreed in advance. With the tradition established, I'll now take a closer look at the mentioned racket sports.
In modern table tennis, scoring up to 11 means a deuce occurs at 10-10. Service alternates every two points and noting that table tennis is point-a-rally, this means that once deuce is reached, each player has served an equal number of times. Service then proceeds to alternate every point with a win-by-two rule, continuing to ensure each player will get an equal number of serves by the end of the game. Therefore, scoring up to an odd number is a part of ensuring that in each game, serve advantage is effectively factored out. Under the pre-2001 system of alternating service every 5 points, the same could be achieved with games up to 11, 16, 21, 26, etc., with 21 being the chosen number possibly due to already established tradition.
In squash and badminton, the server is the player who won the last rally, so there is no similar gameplay rationale. While the original rationale for their scoring systems is probably lost to history, they do in fact have a common origin. This is probably close-court racquets, which is a precursor to squash. Only the server could win points (or "aces"), with the scoring system described in The Book of Racquets (John Ringwood Atkins, 1872) pgs. 44-45:
We have already seen that the open-court game is made with eleven aces; that of the close-court counts fifteen aces. If both players get "thirteen all," the game may be "set" at five aces, or in other words, five additional aces may be added to the score; but if they both reach "fourteen all," then the "set" is usually three aces only. But the "set" is optional.
In the US, this system was maintained in squash for a while, though switching to point-a-rally in 1911 according to Squash: A History of the Game (James Zug, 2007) pg. 30. Outside North America, squash dropped to 9 points in 1926 with only the server eligible to win points (a better reference for this might be in pgs. 39-40 of Zug's book, but those pages aren't free to preview).
In badminton, the earliest available rules from 1873 also uses this exact system. Given the timeline and the rules being written by English servicemen stationed in India, it's highly likely they simply borrowed the system from the racquets/squash played back home. In 1907, the women's singles game dropped to 11 points. All that said, a now lost booklet entitled "Badminton Battledore - a new game" was published in 1860, so we can't entirely rule out badminton as the source of this scoring system. It would seem unlikely though that the better established racquets would need to borrow a system from the newly developing badminton.
At this point and due to its popularity, I'm also going to include volleyball to strengthen the tradition argument, though I can't really find anything too informative. Per Wikipedia, after a brief period of initially being played by baseball-style inning, it was played up to 21 points, dropping to 15 in 1917.
Best I can tell, after squash's 1926 change, none of table tennis, squash, badminton and volleyball then made any major changes to their scoring systems until the late 90s/early 2000s, when they all did with the aim of making the sports more spectator and TV friendly. This makes for 7 decades of these sports being consistently played up to 9, 11, 15 or 21 points. While the original reasons may be obscure (with partial exception of table tennis), I would argue that this has solidified the already existing tradition, though I could not find actual documentation of any specific rationale about the specific numbers chosen.
