The cockatrice was a creature with the body of a rooster and a tail like a lizard, hatched from an egg like a bird but incubated by a reptile. The cockatrice of medieval and Renaissance European legend was thought to have magical powers to kill people with a glance, by touching them, or by breathing on them.
Prior to the quote you cite, Juliet has been longing for the arrival of night and her beloved Romeo - "Come, gentle night, come, loving, black-brow'd night, Give me my Romeo..."
When her nurse enters, Juliet begs for news about Romeo. The nurse, distraught with the news she bears that Juliet's cousin Tybalt has been killed by Romeo, is unable to relay this information immediately and clearly. Juliet interprets her first words to mean that Romeo is dead and begs for a short and simple confirmation of this terrible news, even though she says such a message will affect her as severely as would a glance from a cockatrice.
Hath Romeo slain himself? say thou but 'I,' and that bare vowel 'I' shall poison more than the death-darting eye of a cockatrice.