From the novel "Catch-22", our hero Yossarian was confined in the hospital ward. Officers so disposed were forced to censor the letters of airmen written home. Yossarian was very bored by this and devised games to make it more interesting. He would arbitrarily block out words and letters. "Unspringing" is a poetic device, so I assume the author was using this sarcastically to refer to Yossarian's altering of the letters.

To unspring something is to let it loose as if it was released by a spring. It is not an actual word, but most people will understand it. In the excerpt that you give, the main character, Yossarian, is given the job of censoring letters from servicemen to their families. He has made a game out of this boring job and is trying to see or create patterns by his censorship. By blacking out some words, he can change the meaning and the rhythm of the letters. So by 'unspringing rhythms', he was using his blackout censorship pen to unleash, release new rhythms in the letters, rhythms and meanings that were not intended by the servicemen that wrote them. Hope I didn't confuse you.

Great novel. A little bit difficult to understand without a historical or cultural context though. Randy and FD have done a good job explaining. Next thing you might ask is, "What is a Catch-22?"