we tend to use '잡아먹히다' for an animal being eaten in their perspective. Besides, there is a Korean cuisine called '토끼탕' so this phrase is reasonable in Korea. Never tried one before though.
I had a rabbit sandwich at a restaurant in Portland, Maine recently.
Isn't 네 means mine?
내 - mine; 네 (pronunced [니]) - yours
내: 나+ㅣ(anxient form of possesive)
Literally 네 pronounce [네] but mostly pronounced [니].