I grew up in a conservative, Protestant family and I might be able to shed a little bit of light on this. A few points, simply based on the environment I grew up in, which means of course this is just one perspective on just one religious denomination - I won't provide more caveats than that:
- Christians take the afterlife very seriously. One must be judged by God. Regardless of how confident you are in your final destination, the prospect of being called to account for your actions in front of an omnipotent being is not something they take lightly.
- Protestant Christianity teaches that one must be "saved", that is, accept Jesus Christ and have a profound spiritual experience and subsequent spiritual connection with God. Many Christians are honest people who may not have had such an experience and are aware of this fact, which naturally creates fear when the end of life approaches. In other words, they may have doubts about whether or not they are indeed saved. The alternative to being saved of course, is spending an eternity in hell. As you might imagine, this can be nerve-wracking.
- Christians believe that whether or not someone ought to live or die is in God's hands. They take the view that whatever humans ought to do to extend life ought to be done, since death will come when God wills it. Choosing to stop life support is tantamount to choosing to end someone's life, a power they believe only God wields righteously.
- Lastly, regardless of one's personal or religious beliefs, death is scary. I think the appropriate reaction to a Christian (or any other religious person) who is terrified that they, or their loved one, may die, is not observing that this fear may be incompatible with their professed beliefs. The appropriate reaction is compassion.
Note: I'm an atheist, so telling me that all or part of the above is irrational is...uh...preaching to the choir.