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In this paper we compute the non-abelian tensor square for the free 2-Engel group of rank n > 3. The non-abelian tensor square for this group is a direct product of a free abelian group and a nilpotent group of class 2 whose derived subgroup has exponent 3. We also compute the non-abelian tensor square for one of the group's finite homomorphic images, namely, the Burnside group of rank n and exponent 3.doi:10.1017/s0013091502000998 fatcat:5lmlnq4yubc5zmresov6wtucuq