How we cite our quotes: (Act.Scene.Line) Line numbers correspond to the Riverside edition.
Quote #1
LEONATO
You will never run mad, niece.
BEATRICE
No, not till a hot January. (1.1.91-92)
"Run mad" here refers to catching what Beatrice calls "the Benedick"—essentially going crazy for love. For Beatrice to "catch the Benedick," she’d have to be in love, which she says is as likely as a hot January. That is, "when hell freezes over." Famous last words.
Quote #2
BENEDICK
Then is courtesy a turncoat. But it is certain
I am loved of all ladies, only you excepted; and
I would I could find in my heart that I had not a
hard heart, for truly I love none.
BEATRICE
A dear happiness to women! They would
else have been troubled with a pernicious suitor. I
thank God and my cold blood, I am of your humor
for that. I had rather hear my dog bark at a crow
than a man swear he loves me. (1.1.122-130)
Note that both Beatrice and Benedick say they’ll never love anyone. This little spar is likely saying "of course we’ll never love each other." Invulnerability to falling in love is a point of pride for them both.
Quote #3
BENEDICK
That I neither feel how she should be loved
nor know how she should be worthy is the opinion
that fire cannot melt out of me. I will die in it at the
stake.
DON PEDRO
Thou wast ever an obstinate heretic in the
despite of beauty. (1.1.226-231)
Benedick hates love, and he’s known for hating beauty. Beauty can’t move him to love, but perhaps some more meaningful trait in a girl (like how witty she is, and if she’s named Beatrice) could move him.