When they meet again at Hannah Tupper's, what do Nat and Kit discover that they have in common in The Witch of Blackbird Pond?

Hannah says she first met both
Nat and Kit crying in the meadow. 

Kit and Nathaniel Eaton are both good friends of Hannah Tupper, the supposed witch of
Blackbird Pond. They do not realize this at first. Kit finds the meadow on her own, and it is
the first place she has felt comfortable in Wethersfield. 

Coming home through the meadow everything had seemed so simple, and here it was all
tangled again. Only one thing was sure. She had found a secret place, a place of freedom and
clear sunlight and peace. (Ch. 10) 

Kit befriends Hannah
even though the people of Wethersfield think she is a witch. Kit is amazed that the Quaker woman
does not go to meetings (she has to pay fines), and lives by herself out by the pond in the
swampy land. 

When Nat shows up, Hannah says she should have known that the
two of them would find each other. They have a lot in common. One thing she says they have in
common is that she met both of them the same way, in the meadow.

"'Tis a strange thing, that the only friends I have I found in the same way, lying
flat in the meadows, crying as though their hearts would break." (Ch. 10) 

Nat protests that he was only eight years old at the time. He is
surprised that Kit found Hannah, since her aunt and uncle would not approve. He is definitely
glad to see Kit. 

The friendship between Kit and Nat grows through helping
Hannah. Kit does not realize at first that Nat likes her. She thinks that she should just marry
a local man, William, and be a good Puritan. However, she does not fit in. 

"I can still see the green feathers if I look hard enough. But
they've done their best to make you into a sparrow, haven't they?"

"It's these Puritans," Kit sighed. "I'll never understand them. Why do
they want life to be so solemn? I believe they actually enjoy it more that way." (Ch.
12) 

Later, when Nat illuminates the house William is
building and is banished from Wethersfield, Kit realizes that he really cares about her and she
cares about him. Eventually, she marries Nat instead of William. It is definitely a better
choice for her, since he is the one person who understands her.