What an excellent question! The birds seem to be the imagery most frequently connected to Juliet--she wishes she had a falconer's voice to call Romeo back, and she says she wishes he were like a wanton's bird (a bird on a string) that she could set free but pull back at will. When she initially refers to birds, it seems to be about her desire for power: she wishes she had the power to control Romeo's movements.
As things progress with Romeo, Juliet's imagery becomes increasingly confused: he is first white snow on a raven's back, then a raven disguised as a dove.
Black itself is both friend and foe. Finally, birds signify danger: Paris is a man with the eye of an eagle.
Others apply the bird imagery to her, and to some extent the meaning varies depending on who speaks it. Romeo speaks of her as a "bright angel" and, in response to her falcon metaphor, calls her his "nyas" (that is a bird who has not yet flown or left the nest). When the nurse applies the metaphor to her, it is, in keeping with the nurses' personality, loaded with graphic sexual connotations: she says she must "fetch a ladder by the which your love must climb a bird's nest soon when it is dark." Here the bird's nest is a sexually charged reference not only to the bedroom and but also to the female anatomy, Juliet's.
There is constant tension in the play between the earth and the air, the limits of gravity and the events that can transpire when using Love's light wings. Many of Juliet's references reveal her desire. Romeo's rarefied images reveal his fancy and her virtue. The nurses' imagery, oddly enough, reveal the base, earthy world that the two lovers actually live in.
In Act III, Scene 5, Romeo and Juliet have just spent their honeymoon night together. When they awaken, they hear a bird's song and discuss whether the bird is a lark or a nightingale. The lark would mean it's morning, and Romeo must leave so he won't be caught in Verona, which could mean his death. The nightingale's song would mean it's still nighttime, and the two lovers would have more time together. When Juliet admits it is indeed the lark, she rushes Romeo to leave.
Shakespeare includes many references to birds (nightingales, falcons, doves, swans, crows, etc). Birds are earthly creatures that have the ability to fly free in the sky. They are not obligated to stay down on the ground. Romeo and Juliet wish their love was free like the birds. They too are creatures of the earth, but wish their souls could fly free.
Romeo says, “With love’s light wings did I o’erperch these walls; / For stony limits cannot hold love out, / And what love can do that dares love attempt; / Therefore thy kinsmen are no stop to me.” (II,ii) Romeo will not let Juliet’s family prevent him from reaching his love. Love’s light wings lifted him to Juliet’s bedroom. The Nurse also refers to Juliet’s bedroom as a ‘bird’s nest.”