The concept of an eye for an eye means that punishments for crimes should fit the crime. In early law codes, such as the Mesopotamian tradition from which much of the Old Testament emerged, this was expressed in reciprocal terms. In reality, at least in terms of civil cases (the kind we would call torts today) it may have simply been intended to set the terms by which negotiations could occur, sort of like the concept of the wergeld among Northern Europeans.
As to why Christians would fight in wars, there is a wide diversity of opinion among Christians. Many oppose war based on religious convictions, others do not, as long as they perceive the aims of the war to be just. For others, their definition of Christianity is so entwined with their understanding of patriotism that there is no conflict in supporting any war that their political leadership deems necessary.