Yes, these lines are spoken by the nurse, after she finds Juliet "dead".
The day is black because Juliet, who she loves, is dead. Her light (soul) is turned out (in death) and the, day, therefore is black (and full of darkness, spiritual and emotional). It is not entirely true that the nurse has never seen a day so black, for she has lost her own daughter, her flesh and blood, at an early age; when she says these words, therefore, she is very much in character. This is not to suggest that she is not sad at Juliet's loss--among her household, the nurse is probably the one person who is closest to her. Perhaps this is the worst day of the nurse's life. If so, then even her grief is so verbose that she risks becoming a caricature of herself.