= Solution
For $\sigma=1$, the formal outer family is
$$
z_0=A\exp\left(-\frac1{\log x}\right).
$$
Every nonzero member diverges as $x\to1^-$, so bounded matching selects the outer solution $z_0=0$. Boundary layers are now required at both endpoints.
The $x=1$ layer still has width $O(\varepsilon)$ and leading profile
$$
Z_R(X)=e^{-2X},
\qquad X=\frac{1-x}{\varepsilon}.
$$
Near $x=0$, diffusion and the logarithmically vanishing convection balance on the thinner scale
$$
x|\log x|=O(\varepsilon).
$$
At leading order the reaction term is smaller there. Using the supplied first integral with $\mu=\varepsilon^2$, define
$$
S(x)=x^2(2\log^2x-2\log x+1),
\qquad
I(x)=\int_0^x e^{-S(\xi)/\varepsilon^2}\,d\xi.
$$
The left layer that equals $1$ at the endpoint and matches zero is
$$
Z_L(x)=1-\frac{I(x)}{I(1)}.
$$
Consequently a leading composite description is
$$
\boxed{
z(x;\varepsilon)\sim
1-\frac{I(x)}{I(1)}
+\exp\left[-\frac{2(1-x)}{\varepsilon}\right]
}.
$$
Its sketch has value $1$ at each endpoint, drops sharply to an almost-zero outer plateau just to the right of $x=0$, and rises through an $O(\varepsilon)$ layer just before $x=1$.
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