The Tuck family is different from every other family that Winnie Foster will ever meet, because the Tuck family is a family of immortals. That's right, they will live forever. Makes me wonder if Twilight took any inspiration from this book. After all, that book was about a "regular," loving family of immortals too.
Eventually the author explicitly tells readers that the Tuck family is a family of immortals, so I assume that your question is asking about early clues that the Tuck family is more than a family living in the woods.
Chapter 2 introduces the reader to Mr. and Mrs. Tuck. Mae wakes Tuck up from a peaceful sleep, and he says the following line:
"Why'd you have to wake me up?" he sighed. "I was having that dream again, the good one where we're all in heaven and never heard of Treegap."
That seems like a very odd thing to say. Earlier Treegap was described as a peaceful little hamlet. It hardly sounded like a place people would flee from. The heaven comment is weird too. Essentially, Tuck is wishing he were dead and in heaven.
Mae Tuck tells Tuck that he should get over having that dream because nothing is going to change.
"But, all the same, you should've got used to things by now."
Got used to what? Why won't things change? In fact, what things won't change? It's a very odd conversation that they are having and more than hints to readers that there is something different about the Tucks.
Chapter two's closing line drops the big hint that the Tuck family is not normal.
For Mae Tuck, and her husband, and Miles and Jesse, too, had all looked exactly the same for eighty-seven years.
Wait, what? 87 years! How does anybody look the same for that long of time? I mean, some Hollywood stars definitely try, but Treegap doesn't seem like a place for high priced plastic surgeons. From this point forward, the reader quickly learns about the spring water and its ability to give immortality, and Winnie must decide whether or not she would like to join the Tuck family in their uniqueness.