= Solution
Write $y(x)=Y(X)$ with $X=(1-x)/\sqrt\epsilon$. The transformed equation is
$$
Y''=Y-(1-\sqrt\epsilon X)^2+\epsilon(2+Y^2).
$$
Since the rescaled forcing contains $\sqrt\epsilon$, an <inner expansion> naturally uses powers of $\sqrt\epsilon$. Its leading term satisfies
$$
Y_0''=Y_0-1,\qquad Y_0(0)=0.
$$
The <asymptotic matching condition> with the leading <outer solution> is
$$
Y_0(X)\longrightarrow y_0(1)=1\qquad(X\to\infty).
$$
The general solution is $Y_0=1+Ae^X+Be^{-X}$. Matching eliminates $Ae^X$, and the boundary condition gives $B=-1$. Hence
$$
\boxed{Y_0(X)=1-e^{-X},\qquad
y_{\rm inner}(x)\sim1-\exp\left(-\frac{1-x}{\sqrt\epsilon}\right).}
$$
This tends to the boundary value zero at $x=1$ and to the matching value one on the interior side of the layer. The negative sign in the exponential follows from measuring the <inner variable> inward from the right endpoint.
For an optional uniform leading approximation, combine the leading inner and outer terms and subtract their common overlap value one. The <additive composite expansion> is
$$
\boxed{y_{\rm comp}(x)=x^2-\exp\left(-\frac{1-x}{\sqrt\epsilon}\right).}
$$
It satisfies the right boundary condition exactly; its left-end discrepancy is the exponentially small value $-e^{-1/\sqrt\epsilon}$. The requested leading <inner expansion> is $Y_0$ above.
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