= Solution
Choose the separated cosine solutions instead:
$$
\boxed{\Phi(x,y)=\cos(mx)\cos(py).}
$$
Its derivatives are $\Phi_x=-m\sin(mx)\cos(py)$ and $\Phi_y=-p\cos(mx)\sin(py)$. They vanish on the appropriate opposite sides of the square for integer $m,p$, so the outward normal derivative is zero everywhere on the boundary. Also $\Delta\Phi=-(m^2+p^2)\Phi$.
This is nonzero for every pair of integer indices, including a zero index. For $m=p=0$ it is the constant solution, illustrating the zero mode allowed by <Neumann boundary conditions>. Negative indices give the same cosine modes. The construction is the Neumann part of the <Dirichlet and Neumann spectra of a square>.
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