= Solution
A <permutation> $\pi$ of <identical particles> multiplies a fermionic state by $\operatorname{sgn}(\pi)$; a bosonic state is invariant. Particle exchange keeps $y_+$ fixed and sends $y_-$ to $-y_-$. Consequently the spatial <wave function> is even in $y_-$ for the stated bosonic model and odd for the stated fermionic model. This assumes no additional spin variables in the supplied model; exchange of full particle states is the underlying rule.
The coordinate change is orthogonal, with unit <Jacobian determinant>, so the <kinetic energy> becomes
$$
H=-\frac{\hbar^2}{2m}(\partial_{y_+}^2+\partial_{y_-}^2)
$$
on the square $|y_+|,|y_-|<a$ with zero boundary values. Product states $\psi_r(y_+)\psi_s(y_-)$ have energy
$$
E_{r,s}=K(r^2+s^2),\qquad K=\frac{\pi^2\hbar^2}{8ma^2}.
$$
The one-dimensional <eigenfunction> is even for odd $s$ and odd for even $s$. <Bosons> therefore allow odd $s$, whereas <fermions> require even $s$. The lowest bosonic pair is $(r,s)=(1,1)$; its first excited pair is $(2,1)$. The lowest fermionic pair is $(1,2)$. Thus
$$
\boxed{E_{B,0}=2K=\frac{\pi^2\hbar^2}{4ma^2},\qquad
E_{B,1}=E_{F,0}=5K=\frac{5\pi^2\hbar^2}{8ma^2}.}
$$
The printed bosonic ground-energy claim omits the factor $\pi^2/4$; the value above follows directly from the supplied one-dimensional spectrum and <kinetic energy>.
The separation is $|x_1-x_2|=\sqrt2|y_-|$. The normalized $y_+$ factor integrates out. In the fermionic <ground state>, the relative-coordinate density is $a^{-1}\sin^2(\pi y_-/a)$, giving
$$
\langle|y_-|\rangle_F=\frac2a\int_0^a y\sin^2(\pi y/a)\,dy=\frac a2.
$$
For the bosonic <ground state> it is $a^{-1}\cos^2(\pi y_-/(2a))$, giving
$$
\langle|y_-|\rangle_B=\frac2a\int_0^a y\cos^2(\pi y/(2a))\,dy
=a\left(\frac12-\frac2{\pi^2}\right).
$$
Both <integrals> follow by replacing squared trigonometric functions by their constant and cosine parts and integrating once by parts. Hence
$$
\boxed{\langle|x_1-x_2|\rangle_F=\frac a{\sqrt2},\quad
\langle|x_1-x_2|\rangle_B=\sqrt2a\left(\frac12-\frac2{\pi^2}\right),}
$$
and the difference is $\boxed{2\sqrt2a/\pi^2}$. The odd relative state vanishes at coincidence, which increases its expected separation.
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