Solution:

$\begin{array}{l}
\left|\begin{array}{ccc}
b^{2}+c^{2} & a^{2} & a^{2} \\
b^{2} & c^{2}+a^{2} & b^{2} \\
c^{2} & c^{2} & a^{2}+b^{2}
\end{array}\right| \\
=\left|\begin{array}{ccc}
2\left(b^{2}+c^{2}\right) & 2\left(c^{2}+a^{2}\right) & 2\left(a^{2}+b^{2}\right) \\
b^{2} & c^{2}+a^{2} & b^{2} \\
c^{2} & c^{2} & a^{2}+b^{2}
\end{array}\right|\left[R_{1}^{\prime}=R_{1}+R_{2}+R_{3}\right]
\end{array}$
$\begin{array}{l}
=2\left|\begin{array}{ccc}
\left(b^{2}+c^{2}\right) & \left(c^{2}+a^{2}\right) & \left(a^{2}+b^{2}\right) \\
b^{2} & c^{2}+a^{2} & b^{2} \\
c^{2} & c^{2} & a^{2}+b^{2}
\end{array}\right|\left[R_{1}^{\prime}=R_{1} / 2\right] \\
=2\left|\begin{array}{lll}
c^{2} & 0 & a^{2} \\
b^{2} & c^{2}+a^{2} & b^{2} \\
c^{2} & c^{2} & a^{2}+b^{2}
\end{array}\right|\left[R_{1}^{\prime}=R_{1}-R_{2}\right]
\end{array}$
$=2\left[c^{2}\left\{\left(c^{2}+a^{2}\right)\left(a^{2}+b^{2}\right)-b^{2} c^{2}\right\}+0+a^{2}\left\{b^{2} c^{2}-c^{2}\left(c^{2}+a^{2}\right)\right\}\right]$ [expansion by first row]
$\begin{array}{l}
=2\left[c^{2}\left(c^{2} a^{2}+a^{4}+b^{2} c^{2}+a^{2} b^{2}-b^{2} c^{2}\right)+a^{2}\left(b^{2} c^{2}-c^{4}-a^{2} c^{2}\right)\right] \\
=2\left[a^{2} c^{4}+a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} b^{2} c^{2}-a^{2} c^{4}-a^{4} c^{2}\right] \\
=4 a^{2} b^{2} c^{2}
\end{array}$