= Solution
For $\sigma=-1$, the reduced first-order outer equation is
$$
4x\log^2x\,z_0'+4z_0=0.
$$
The condition at $x=0$ selects
$$
\boxed{
z_0(x)=\exp\left(\frac1{\log x}\right)
}.
$$
Indeed $z_0\to1$ as $x\to0^+$, while $z_0\to0$ as $x\to1^-$. The boundary condition at $x=1$ must therefore be supplied by a <boundary layer>.
Set
$$
X=\frac{1-x}{\varepsilon},
\qquad z(x;\varepsilon)=Z(X).
$$
Since $x\log^2x=O(\varepsilon^2X^2)$ in this layer, the convection term is lower order. The leading inner equation and matching conditions are
$$
Z_{XX}-4Z=0,\qquad Z(0)=1,\qquad Z\to0
\quad(X\to\infty).
$$
Thus
$$
\boxed{Z(X)=e^{-2X}}.
$$
The leading uniformly valid expression is
$$
\boxed{
z(x;\varepsilon)\sim
\exp\left(\frac1{\log x}\right)
+\exp\left[-\frac{2(1-x)}{\varepsilon}\right]
}.
$$
Near $x=0$ diffusion regularizes the divergent derivative of the outer approximation on the thinner scale $x|\log x|=O(\varepsilon)$, but the leading value there remains the already matched constant $1$.
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