Jesse does not want Winnie to drink from the spring because she will become immortal.
Winnie accidentally sees Jesse drink from the special spring in the woods outside her family’s house. She just thinks it is an ordinary spring. She has never seen Jesse before and does not understand what she is seeing.
Jesse and the rest of the Tuck family drank from the spring when they stopped at the farm eighty-seven years before. After a while they realized that they were all immortal. They could not be injured and they could not die.
Winnie knows nothing about this, of course. She is just wandering around because she is bored. She is excited to see another kid, and thinks it is funny when he tells her that he is one hundred and four years old. He then clarifies that he is seventeen, but he really is that old. He just has not aged in over eighty years.
When Winnie wants to drink from the spring, Jesse tries to stop her.
Jesse Tuck's face was instantly serious. "Oh, that. No—no, it's not," he said quickly. "You mustn't drink from it. Comes right up out of the ground. Probably pretty dirty." And he began to pile the pebbles over it again. (Ch. 5)
She asks him why he drank it, and he realizes that she saw him. He does not want her to make the same mistake that his family did. Since she is persistent he and his mother end up taking Winnie home so that they can explain in detail.
Later, Jesse does ask Winnie to drink the water. He wants he to wait until she is his age, and then drink it. They will then both be immortal and they can live together and possibly get married. Winnie considers the offer, but eventually does not accept it.