Why is Connor being unwound?

Readers are never told exactly why Connor is being unwound, but chapter 1 does indicate that Connor is a bit of a troublemaker. His grades aren't fantastic. In fact, the best grade he's ever gotten in science is a B+. He has anger and temper issues, and he lets his anger control many of his actions. Readers come to learn that Connor is no stranger to school fights.

For once he had held his temper, and aside from a few fights in school that weren't his fault, he kept his emotions hidden.

We are also specifically told that Connor's defiant behavior has resulted in him being in and out of various disciplinary schools, so readers know that his misbehavior is more extreme than the occasional teenage rebellion.

They think that just because he's been in and out of disciplinary school they have a right to judge him.

In any normal society, Connor's parents would be forced to deal with him; however, in this particular dystopia, parents of teenagers have a different option. They can retroactively abort their child. This process is called unwinding, and all parts of an Unwind's body "live on" with the recipient of each body part.

In the dysoptian world in which Unwind is set, errant children and young adults between the ages of 13 and 18 can be sentenced to be "unwound." This means that they can be taken to what are called harvest camps where their organs can be removed and harvested for use at a later date. Though initially devised as a compromise on the issue of abortion that led to the Second Civil War, the practise of unwinding has become a method of control, a way of keeping rebellious young people in check.
Connor is one of those young people. He's a bit of a trouble-maker, always getting into fights. His parents are at their wits' ends and don't know what to do with him. In the terrifying dystopian vision of the United States presented in Unwind, troublesome youngsters aren't sent to boot camp; they go to harvest camps. These camps don't just serve as a method of social control; they're also a resource for parents who either can't handle their kids or don't want to.