When Sal's father tells her that her mother won't be coming back, she's initially in a state of denial. She simply can't believe that her mother won't return home.
As "proof" that she must be coming back, Sal heads off to her room and brings back all the postcards that her mother has written to her. If her mother has written all these postcards, she figures, then she must be coming home and some point. Otherwise, why else would she bother writing them? Clearly, she wants to maintain some kind of connection with her daughter, and so the next logical step would be for her to come home.
For good measure, Sal goes out to the chicken coop and brings her mom's favorite chicken. Why would she leave this bird behind? She loves this chicken.
But in retrospect, Sal realizes that what she really meant by this was
How can she not come home to me? She loves me.
In other words, it's not about the chicken; it's about Sal. She feels terrified at the prospect of her mother not coming home, and all this talk about postcards and chickens is just a way for her to avoid the terrible truth.
Phoebe did exactly the same when she brandished her mother's sweater in front of her father. She too couldn't deal with the truth that her mother wasn't coming back and so transferred her feelings to an object as a way of minimizing how badly she must have felt inside.