Using the possessive here sounds much more natural. To use the other expression, you need to say "those of," because it's referring to eggs, which is plural.
I don't want to put my eggs in a fridge like those of a chicken.
This is grammatical and understandable, but sounds a little stilted. Since your question was "when to prefer" this usage, I'm trying to think of a time when I might prefer it, but my imagination is failing me. The only thing I can think of is if I were trying to write a story with an extremely proper or curmudgeonly character in it, I might actually want her speech to sound like this.