 Solution

In both collinear trial states, a spin and its second neighbor are parallel. The next-nearest-neighbor energy is therefore $-J_2{S}^2$ per spin in either case. Making $J_2$ negative makes that contribution positive and raises both energies.

A negative $J_2$ prefers opposite second-neighbor spins. Neither all-parallel order nor the simple alternating nearest-neighbor pattern satisfies that preference: in the latter, alternating twice returns to the original direction. This competition is a source of frustration when the nearest-neighbor interaction favors one of those collinear patterns, and allows a noncollinear twist to lower the energy. Merely calling $J_2$ antiferromagnetic does not mean that the nearest-neighbor alternating state satisfies it.