= Solution
Place the four fixed charges at $(sa,ta)$ with $s,t\in\{-1,1\}$, and write the moving particle's position as $(x,y)$. Introduce its mass $m>0$, which is needed for an oscillation frequency but is not named in the question. Let $C=Qq/(4\pi\epsilon_0)>0$. By <Coulomb's law> and superposition, the moving particle's <potential energy> is
$$
V(x,y)=C\sum_{s,t=\pm1}\left[2a^2-2a(sx+ty)+x^2+y^2\right]^{-1/2}.
$$
The <electrostatic potential> itself is $V/q$. Put $r^2=x^2+y^2$ and $\delta_{st}=-(sx+ty)/a+r^2/(2a^2)$. Near the centre, a <Taylor expansion> gives
$$
\left[2a^2-2a(sx+ty)+r^2\right]^{-1/2}=\frac1{\sqrt2a}\left(1-\frac12\delta_{st}+\frac38\delta_{st}^2+\cdots\right).
$$
On summing over the four corners, linear terms cancel. To quadratic order, $\sum\delta_{st}=2r^2/a^2$ and $\sum\delta_{st}^2=4r^2/a^2+O(r^4/a^4)$, since $\sum st=0$. Reflection symmetry removes all odd-degree terms. Hence
$$
V(x,y)=\frac{C}{\sqrt2a}\left(4+\frac{x^2+y^2}{2a^2}\right)+O\left(\frac{Cr^4}{a^5}\right)=V(0,0)+\frac12\kappa(x^2+y^2)+O(r^4),
$$
where
$$
\kappa=\frac{C}{\sqrt2a^3}>0.
$$
Thus the centre has zero planar force and a positive-definite planar <Hessian matrix> $\kappa I_2$. It is a strict local minimum of the constrained <potential energy>, so it is a <stable equilibrium>. More explicitly, conservation of the total energy confines sufficiently small-energy motions within a small neighbourhood of this isolated local minimum.
Keeping only the quadratic potential, the two coordinates obey independent <small oscillations>:
$$
m\ddot x=-\kappa x,\qquad m\ddot y=-\kappa y.
$$
Both planar modes have the same <angular frequency>, and the ordinary <frequency> is
$$
\boxed{\omega=\sqrt{\frac{Qq}{4\pi\epsilon_0\sqrt2\,ma^3}},\qquad \nu=\frac{\omega}{2\pi}.}
$$
The planar constraint is essential to this <constrained electrostatic equilibrium>. Along the perpendicular axis, $V(0,0,z)=4C/(2a^2+z^2)^{1/2}=V(0,0)-Cz^2/(\sqrt2a^3)+O(z^4)$, so the same stationary point is unstable to unconstrained normal displacements. That also checks the cancellation of the three-dimensional Hessian trace required by <Laplace's equation> away from charges; it does not contradict the positive-definite restriction to the given plane.
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