When Juliet's nurse first tells her that Tybalt is dead, the nurse does not actually say right away who has been killed, leading Juliet to believe that she is talking about Romeo. When Juliet asks if Romeo has "slain himself," she requests a one-syllable response—aye or nay—and, yet, the nurse only describes the "bloody piteous corse" that made her swoon (3.2.51, 60). Assuming that the nurse is describing Romeo's dead body, Juliet says,
O break, my heart, poor bankrout, break at once!
To prison, eyes; ne'er look on liberty.
Vile earth to earth resign; end motion here,
And thou and Romeo press one heavy bier. (3.2.63–66)
Here, Juliet addresses parts of her body because she longs to die rather than live without Romeo. She tells her heart to break, and she tells her eyes to close forever. She commands her body to go into the earth, suggesting that her body and Romeo's can now share one resting place.
Later, when Friar Lawrence tells Romeo that the Prince has sentenced him to banishment rather than death, Romeo reacts in a way that the friar does not expect. Rather than be grateful for his life, Romeo is more upset than if he had been sentenced to death for killing Tybalt. Romeo says,
Ha, banishment? Be merciful, say "death,"
For exile hath far more terror in his look,
Much more than death. (3.3.13–15)
Later, he explains that the reason for this is that "Heaven is here / Where Juliet lives," and he cries that every unworthy animal will get to stay in Verona and look at her, but he will not (3.3.31–32). In other words, then, he is upset about banishment because he would rather die than be separated from Juliet; remaining alive while knowing that she is in the world and they cannot be together would simply be too painful. These two sections of the text show that each of the young lovers would rather die than live without the other.
In the scene that Juliet learns of Romeo killing Tybalt, she expresses that she is more hurt by the fact that Romeo is being banished and says:
Wash they his wounds with tears: mine shall be spent,
When theirs are dry, for Romeo's banishment.
Take up those cords: poor ropes, you are beguiled,
Both you and I; for Romeo is exiled:
He made you for a highway to my bed;
But I, a maid, die maiden-widowed.
Come, cords, come, nurse; I'll to my wedding-bed;
And death, not Romeo, take my maidenhead!
Here Juliet is saying if she can't be with Romeo, she'd rather die than let any other man have her.
Later in scene iii, Romeo confesses willingness to kill himself figuratively:
As if that name,
Shot from the deadly level of a gun,
Did murder her; as that name's cursed hand
Murder'd her kinsman. O, tell me, friar, tell me,
In what vile part of this anatomy
Doth my name lodge? tell me, that I may sack
The hateful mansion.
Drawing his sword
Here, he was asking the friar where and how he should kill himself for the wrong he has done to Juliet by killing her cousin.
Both Romeo and Juliet show their desperation in these scenes. Their willingness to kill themselves shows the passion they must have had to be together.
Both Romeo and Juliet make such a claim to Friar Lawrence after learning of the Prince's sentence of banishment for Romeo's slaying of Tybalt. Of course, banishment spurs the fear that Romeo and Juliet will never be able to see one another again.
In Act III, Romeo states:
Hence-banished is banish'd from the world,
And world's exile is death: then banished,
Is death mis-term'd: calling death banishment,
Thou cutt'st my head off with a golden axe,
And smilest upon the stroke that murders me.
And Juliet, not to be outdone, then questions Friar Lawrence about any possible remedy to Romeo's banishment:
Tell me not, friar, that thou hear'st of this,
Unless thou tell me how I may prevent it:
If, in thy wisdom, thou canst give no help,
Do thou but call my resolution wise,
And with this knife I'll help it presently.
God join'd my heart and Romeo's, thou our hands;
And ere this hand, by thee to Romeo seal'd,
Shall be the label to another deed,
Or my true heart with treacherous revolt
Turn to another, this shall slay them both:
Therefore, out of thy long-experienced time,
Give me some present counsel, or, behold,
'Twixt my extremes and me this bloody knife
Shall play the umpire, arbitrating that
Which the commission of thy years and art
Could to no issue of true honour bring.
Be not so long to speak; I long to die,
If what thou speak'st speak not of remedy.
Both Romeo and Juliet are willing to die rather than suffer through banishment and not being able to be together.
I think that both Romeo and Juliet say things in this act that show that they would rather die than be separated. For Romeo, there is a good quote in the lines he speaks to Friar Lawrence in Act III, Scene 3. He says
Ha, banishment! be merciful, say 'death;'
For exile hath more terror in his look,
Much more than death: do not say 'banishment.'
To me, what he's saying is he'd rather die than be exiled. Juliet seems to feel the same way. In Act III, Scene 2, she says to the nurse (when she thinks Romeo, and not Tybalt, is dead)
O, break, my heart! poor bankrupt, break at once!
To prison, eyes, ne'er look on liberty!
Vile earth, to earth resign; end motion here;
And thou and Romeo press one heavy bier!
To me, this means that she wants to be buried with Romeo in one coffin (bier).