Why was a newspaper crossword puzzle used to find prospective code-breakers?

I suspect the main issue here is that the crossword in question was a British-style cryptic crossword, whereas your opening paragraph suggests you are imagining an American-style newspaper crossword.

If you read books that deal with the British code breakers, they seemed to have various criteria for finding people; for example, they wanted "upper class" females because it was felt they could keep a secret. At the same time, they found people who by almost any measure were very bright, such as Turing and Alexander (who was the best chess player in Britain and also a mathematician). Golombek was another top chess player and there were other talented mathematicians like Jack Good (who won the Smith's Prize which if you look it up has some amazing winners).

As for crossword puzzle solvers: Firstly, even with brilliant leadership there turned out to be a tremendous amount of grunt work required that probably it was felt that x-word solvers would be tolerant of. Secondly, some crossword puzzles are pretty challenging but as you indicate, maybe don't prove that the person is brilliant. Probably they recruited people who said they liked puzzles, as even want ads today specify, and interviewed from among them. Note that it was Turing who developed ways of automating the process but when the project started they needed human "computers" (and needed even after automation) which are people who are good at and willing to do routine detailed tasks that today are performed by electronic computers but the term originally meant a human who did basic math by hand.

The Telegraph explains:

“Whether it’s a simple cipher, or something as complex as the codes of the Enigma machine which the Bletchley codebreakers were working on, the trick is making links between letters and words,” says Michael Smith, the author of Station X: The Codebreakers of Bletchley Park (and who interviewed Mr Sedgewick for the Telegraph in 1998). “Crosswords are the same sort of lateral-thinking exercise.”

Just as with crosswords, where working out 15 Down gives you a few letters in the Across clues, you can use the information from cracking part of a code to crack the rest of it; it is, says Smith, a very similar logical procedure.

But more importantly, crosswords are about getting inside the mind of your opponent, and in the same way, codebreaking was about getting inside the mind of your enemy. The codebreakers came to know the people encoding the messages individually, by their styles, as crossword-solvers come to know setters. One, Mavis Batey, worked out that two of the Enigma machine operators had girlfriends called Rosa: “She worked it out, trying different options, like in a crossword. Once it worked once, it was an obvious option elsewhere,” says Smith.

Note that this article also contains the crossword used to recruit the Bletchley codebreakers.

There are excellent answers above but I would still like to add a couple of excerpts from the book The Code Book by Simon Singh:

• "As crossword addicts know, when a word is partly completed it is often possible to guess the remainder of the word.": Educated guessing is an important part of code breaking and who's better than crossword addicts in filling up gaps?

• "It was felt that crossword experts might also be good codebreakers, complementing the scientific minds that were already at Bletchley...": I guess this is self explanatory.

Edit:

This is a screenshot of the actual crossword in question, courtesy the eBook version of the above book, which credits it to The Daily Telegraph. If you solve (a majority of) it in under 12 minutes, then congratulations, you could have helped crack the Enigma.