Another reason is that if n did not equal 2, some of the symmetries of the Newton equation would be lost. For example, classically, physics on a microscopic scale is time reversal invariant. We can see this from the Newton equation because if x(t) is sent to x(-t), the 2nd time derivative ensures that the negatives cancel. If n was an odd number, we would not observe this symmetry.
If n was smaller than 2, then Galilean transformations would not leave the equation invariant. If we send x(t) to x(t) + vt, then the second time derivative kills the vt term on the LHS. If n=1, then this would not be possible, and relative velocity would be meaningless (even non-relativistically). If n was some number larger than 2, then transformations that send x(t) to x(t) + b(t^m), where m is less than n, would be a symmetry of the Newton equation. But relative accelerations, jerks, etc. produce observable discrepancies, so that should not be possible either.
Again, this may not really answer "why" in the sense that you are asking, but the fact that the equation matches observations is enough in science to justify it. The Newton equation is basically experimental fact, like Vladimir says.