The past really seems to fly out the window in the alien world of the prison camp. Everyone is a prisoner now after all, and a prison sentence is a sort of great social equalizer, impacting rich and poor, young and old. But what people were in the outside world definitely effects the type of prisoners they become, as well as their chances for survival. The ex-Captain Buynovsky is used to giving orders and can't adjust to taking them in the camp; the ex-carpenter Shukhov carves out a niche for himself as a skilled laborer.
In fact, the past might dictate a person's future in the camps more than it ordinarily would in the outside world. For a prisoner in One Day in the Life of Ivan Denisovich, the present is sort of frozen in the camps, and the future is put indefinitely on hold since prison sentences are practically never-ending. In a way, the past becomes the main source of a person's identity in the camp. It's no mistake that Shukhov often describes people by what they were in the outside world. Buynovsky is still "the Captain," even if he technically isn't anymore. The camp disconnects people from their past, but the past still remains an important factor in daily camp life.
Shukhov's past, and the other characters' pasts, provide interesting contrast to their present lives, but the pasts ultimately do little to define who they are as prisoners in the present.
For all of Gang 104, the past remains hugely important since a unique, individual past is one of the only ways to distinguish the mass of zeks from one another.