= Solution
For $\Phi=X(x)Y(y)$, <separation of variables> gives $X''/X+Y''/Y=-(m^2+p^2)$. With the standard separation constants $m^2$ and $p^2$, the zero-endpoint solutions are sines. Thus, provided $mp\ne0$,
$$
\boxed{\Phi(x,y)=\sin(mx)\sin(py)}
$$
is nonzero, has zero value on all four sides, and satisfies $\Delta\Phi=-(m^2+p^2)\Phi$.
There is a missing qualification in the printed request for arbitrary integer $m,p$. The <Dirichlet and Neumann spectra of a square> show that a nonzero Dirichlet solution exists precisely when
$$
m^2+p^2=j^2+k^2\quad\text{for some integers }j,k\geq1.
$$
When one of $m,p$ is zero, the displayed sine product vanishes identically, and a nonzero solution need not exist: for example $(m,p)=(0,0)$ or $(0,1)$ admits none. On the other hand $(0,5)$ does admit $\sin(3x)\sin(4y)$. Thus the natural intended case is two nonzero integer indices, with the stated spectral criterion covering the exceptional cases.
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