It is our belief that Christ, the one person with two natures, died on the cross. As for His nature… a nature cannot die, because it is not a person. A rabbit may be dying, but the fact of what it
is - “rabbithood” or the rabbit’s nature - is not being harmed or destroyed. When the rabbit is dead, it’s the whole rabbit that’s dead, nature and all.
On the other hand, Christ had the power to die and then be raised again by His own power, which is spoken of by the early Church fathers as a sign of baptism. I’ll paraphrase a bit from St. Cyril of Jerusalem: Christ had the power to go down into the earth, and to rise again. We now imitate that action by going down into the earth, from which we have the power to rise again. We all just do what we can.