The quote "Never seen so black a day as this.  Oh, woeful day, oh woeful day" was said by the Nurse in Act 4, Scene 5 when they found Juliet in her bed and assumed she had committed suicide.  She had taken a potion the Friar gave her to make her seem that she was dead.

The quote means that there has never been a sadder day due to the death of Juliet.  This is a day to mourn because her death has blackened the day.

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.

It is the Nurse who speaks these lines in 4.45.49-54. She is reacting to the news that Juliet (appears) to be dead.

Here are the lines in question:

O woe! O woeful, woeful, woeful day
That ever, ever I did yet behold.
O day, O day, O day, O hateful day.
Never was seen so black a day as this.
O woeful day, O woeful day.