In Chapter 5 of Natalie Babbit's Tuck Everlasting, Jesse is glad to see his mother Mae and older brother Miles arrive at the spring because Jesse wants to protect Winnie from the supernatural powers of the spring but is unsure of how to go about it.
In Chapter 5, Jesse becomes very distressed when Winnie witnesses him drinking from the spring and wants a drink too. He tries very hard to dissuade her, even trying to tell her the water is dirty when she asks him if the water in the spring is good to drink, as we see when he says, "No--no, it's not [good to drink] ... You mustn't drink from it. Comes right up out of the ground. Probably pretty dirty." When that fails, he tries dissuading her by telling her it's unhealthy for her and that it would even be "terrible" for her to drink from the spring. When all of his efforts to dissuade her fail, he becomes relieved when he sees his mother and brother returning to the spring, saying to himself, "They'll know what to do."
Later, in Chapter 7, we learn exactly why Jesse felt so desperate to protect Winnie from drinking from the spring. We learn that 87 years ago, the Tuck family drank from the spring and became invincible and immortal. While being invincible and immortal has its problems, it can also be "kind of wonderful," as Jesse phrases, but most importantly, it can also be very dangerous, as Mae warns. A great many terrors and hardships would enter the world if suddenly all of mankind was able to be invincible and immortal. Since Mae understands the dangers, she knows the powers of the spring must be kept a strict secret and whisks Winnie off to the Tucker family home so that her husband can make sure Winnie fully understands the power of the spring and why the secret must be kept. Mae tells Winnie they don't intend to hurt her, but she needs to know the truth.
It pains me to think how your ma and pa will worry, but there's no way around it. We got to take you home with us. That's the plan. Tuck--he'll want to talk it out, make sure you see why you can't tell no one. (Ch. 7)