When Huck is helping Jim, we see his "support" of civilization mainly through how he thinks that slavery is right, that slaves are a lower class of people, and that he is wrong to help a runaway slave.
In chapter 8, Huck says he won't tell on Jim, even though "people would call me a low-down Abolitionist and despise me for keeping mum". Later Huck gets mad at Jim for planning his future as a free man, "coming right out flat-footed and saying he would steal his children-children that belonged to a a man I didn't even know; a man that hadn't ever done me no harm." Huck decides to turn him in right then, but can't, and after "knowed very well I had done wrong." In chapter 31, Huck decides to write Miss Watson and turn Jim in. After writing the letter, he "felt good and all washed clean of sin", but can't do it, and decides that, for Jim, he would "go to hell", tearing the letter up.
All of these quotes show how Huck's attitude towards slavery supports Civilization's attitude at the time. The key is at the time, because we now know that Huck was doing the good thing, and that his conscience was right.