I also think it's false that Skinner really said it (exactly like that), and even if he said it offhand verbally, it's almost certain he didn't put it in writing in that form.
My argument is that the 4-gram "shape him into anything" doesn't show in n-gram viewer, which means there are very few books containing it... according to the help. If you try "give me a child" you'll get an n-gram as there are plenty of books containing that 4-gram.
A book search (instead of n-gram) for the phrase "shape him into anything" returns only 10 books or so. 3 or 4 of these attribute it to Skinner, one reservedly, which is a psychology textbook. The other books are on the margins of psychology (philosophy etc.) so they likely propagate hearsay. None of the books containing that phrase is a biography of Skinner, and there are some book-length biographies of him. Also none of the books say or cite where he might have said or written it.
The exact same result (no n-gram, few books) also applies to the 4-gram "child and I'll shape".
As a related trivia, some of the books that hit "give me a child", have it for "give me a child until he is seven, and he is mine for life," attributed to unnamed Jesuits, but other books cast doubt on veracity of the attribution. I'm mentioning this because one book drew a connection between what Skinner supposedly said and what Jesuits supposedly said.
As for a passage that ideatically comes close to that and we know Skinner wrote... here's one in SEP
Skinner protests that “it is in the nature of an experimental analysis of human behavior that it should strip away the functions previously assigned to autonomous man and transfer them one by one to the controlling environment” (1971, p. 198).
I read a few more pages from Skinner's Science and Human Behavior (195X). The passage I quoted is actually quite representative of Skinner's pretty convoluted writing style. The hypothesis that he would have written anything in a more direct fashion seems pretty unlikely. I don't know how he spoke...