It is clearly described in scripture that
Jesus
did in fact
die
and was buried (Matthew 27:50-61; Mark 15:37-47; Luke 23:44-55; John 19:30-42). Since Jesus is the complete incarnation of God, fully God and fully man (a concept referred to as the
hypostatic union
), this causes us to use terminology that may imply that God Himself died – a conundrum of the highest order. Even Nietzsche's announcement that "God is dead" was not literal, but a proclamation of a shift in intellectual thought that no longer acknowledged God. From the most devout Christian to the Deist who believes that God is not involved in the world to Atheism itself, everyone understands that the concept of an all-powerful, everywhere present, all-knowing (i.e. omnipotent, omnipresent, omniscient) being would inherently include the inability of that being to cease to exist.
It is in the particular verbiage "cease to exist" that we may find the answer to our riddle. All human beings will at some point experience the breakdown of their bodies to the point of bodily "death." However, this is not the end of our – or Jesus' – existence. We are more than just a carbon-based organism, and although God has designed our bodies to be integral to our existence, they are not its definition. Our souls continue
after our bodies die
, and even they will be replaced with glorified bodies at the rapture, when "the dead in Christ will rise first" (1 Thessalonians 4:16). Jesus' death was no different than ours, except that His soul – or the immaterial part of Him – was equally the immaterial part of a human being and God Himself. That part of Him continued on, and was resurrected, just as we will be, into a glorified body.
So, did God die? One of the members of the
Trinity
experienced the death of His body, but did not cease to exist in the ultimate sense. Jesus continues existing in a glorified human body, and will for all eternity.