Why do people always bring God in to discussions like this? Your belief in a God should not allow you to dictate what others can do with their bodies.
We accept that forcing animals to live in constant pain is inhumane, but we don't extend the same reasoning to humans? Makes no sense, in my opinion.
Provided the person in question can prove they are of sound mind, or have a valid advanced directive, they should be allowed to die with some dignity.
The major reason why God is important is this. Nobody knows for a fact that God and His Bible are not the real thing.
If God is real, then it is important to follow what He says.
Got says to repent from unbelief in what He says - Jesus salvation - and from evil. Murder, even if it is self-murder, is evil. If a person commits the evil of self-murder, he might not have enough time to repent if he thinks better of it. So, it becomes an unforgiven sin in the eyes of God... a lasting imperfection. In eternity, the person will be thrown into Hell.
But even if self-murder is not a sin somehow, or is something that God condones in certain circumstances, there is, in the least, plenty of Biblical evidence that it might be totally wrong. Better to be safe than sorry.
Now, if anyone can prove that God doesn't exist, it's a different story. But the fact of the machine-like universe essentially proves that God exists. Machines have makers.
God is important in everything, but especially important in this because of the lasting effects of death. Not believing God doesn't make Him to not exist.