"The chicken is the only animal we eat before it’s born and after it’s dead” is a joke (or saying) that’s been used from vaudeville days. Forms of the joke have been cited in print since at least the 1910s.
February 1915, San Joaquin Light and Power Magazine (Fresno, CA), pg. 105, col. 2:
The Versatile Hen
Fuzzy Bo: “Talk about yer eagles all ye want to, but give me the good American hen first every time.”
Andy: “Yep, de hen’s a wonder.”
Fuzzy Bo: “Sure, it’s a wonder; ye can eat it before it’s born, an’ ye can eat it after it’s dead; nuthin’ on earth can beat that.”
11 March 1920, Lubbock (TX) Avalanche, pg. 6, cols. 3-4:
The teecher ast of Bilsters who was the most wunder full animal in the earth and he sed it was the chicken. She sed Why and he sed it was because it is good to eat before its born and all so after it died.
He: What animal is it that you eat before it’s born and after it’s dead?
She: I’ll bite.
He: A chicken. — Purple Cow
Minstrel Memories: A gay ‘90s minstrel
By Tom Taggart
New York, NY: Samuel French
TAMBO. Good? Dey haven’t laid a bad egg yet. You know, it’s a funny thing about chickens —
INTERLOCUTOR. What is?
TAMBO. It’s de onliest animal you can eat before it’s born.
INTERLOCUTOR. True, quite true.
Backyard Poultry Raising:
The chicken-growing, egg-laying, feather-plucking, incubating, caponizing, finger-licking handbook
By John F. Adams
Garden City, NY: Doubleday
An old vaudeville gag had it that the chicken must be the world’s most useful animal— you can eat it before it’s born and after it’s dead.
The Outlaw Trail
By Robert Redford
New York, NY: Grosset & Dunlap
“Did you know the chicken is the only thing we eat before it’s born and after it’s dead?” said Tex.