The civilized people in Twain's Huck Finn don't have any fun, either.  They have to do things like sit in church and go to school.

On a more serious note, to be civilized in the novel is to have no conscience.  Huck isn't aware of it for most of the novel, but when he befriends Jim he is being uncivilized, according to the home he was raised in and the society he lives in.  He is raised by people that actually own slaves, of course. 

Huck has a conscience.  He thinks that's a negative, but he still follows it most of the time, and rationalizes reasons why he should. 

He never directly condemns slavery as a whole, but he does give in to his conscience and, in the end, helps save Jim.  This is a very uncivilized thing to do. 

The reason Huck fears this is that he does not want to be forced to abide by the rules that "civilized" people are supposed to follow.  He wants to be able to do his own thing and to act in whatever way he feels is right.

There are sort of minors ways he wants to do this.  He wants to be able to dress how he wants and chew tobacco and things like that.

But there are also more important things.  As we see over the course of the book, he disagrees with society on such things as slavery and so he wants to be able to follow his own moral code (not necessarily just about slavery, but on everything in general).