For a polynomial $f$ define $\Delta_c f$ by $(\Delta_c f)(x) : = f(x + c) - f(x)$. This is a polynomial of degree one less than $f$ (or the $0$ polynomial if $f$ was constant). Then consider $\Delta_4 \Delta_2 \Delta_1 f$, it is $0$. Expanding it gives that it is $(x + 7)^2 - (x + 6)^2 - (x + 5)^2 + (x + 4)^2 - (x + 3)^2 + (x + 2)^2 + (x + 1)^2 - x^2 = 0$. This reduces the original problem to checking those cases with $n \leq 7$, which can easily be done. One couls also use $\Delta_2 \Delta_1 f$ to be $4$, easily reducing the problem to $n \leq 3$, making the checking easier. This also shows that one can replace the square by arbitrary powers $k^n$ if one just changes $4$ into a big enough number (the biggest value occuring in the first $2^{n + 1}$ numbers is the best possible bound, a more general bound is $n! \cdot 2^{\frac {n(n - 1)}{2}}$ or $2^{n^2}$). See also http://www.mathlinks.ro/viewtopic.php?t=30998 for a related problem.