Nope, it does not say that. The commandment is "thou shall not kill." True Christians believe only God has the right to end life.
The "eye for an eye" thing is on the old testament. Christians live by Christ's forgiveness and the new testament, which clearly states that killing is wrong. If you really want to go by everything in the old testament, it also talks about incest. You're not for that too, are ya? Job having sex with his daughters after getting drunk with them? Do you REALLY want to go back to the old testament?
Since real Christians believe everyone but Jesus is a sinner, then what gives one sinner the right to end the life of another via the gov't killing them? Ending a life to protect your own is one thing, summarily killing someone with poison in a vein 20 years after their crime is something entirely different.