Fourier transformation of the Toeplitz matrix elements in the infinite-matrix limit gives, $$w(\theta)=\sum_{m=-\infty}^\infty \frac{e^{im\theta}}{1+im\epsilon}=\frac{i}{\epsilon} \left[e^{-i \theta} \Phi \left(e^{-i \theta},1,\frac{\epsilon+i}{\epsilon}\right)-\Phi \left(e^{i \theta},1,-\frac{i}{\epsilon}\right)\right],$$ with $\Phi$ the Hurwitz-Lerch transcendent. The Fourier transformed matrix is diagonal, so the estimate for the eigenvalues of $G$ for $n\gg 1$ is $$\lambda_k=w(2\pi k/n),\;\; k=1,2,\ldots n.$$ This is a monotonically decreasing fnction of $k$. The smallest eigenvalue is reached for $k=n$, $$\lambda_{\rm min}=w(2\pi)=\frac{i}{\epsilon} \left[ \Phi \left(1,1,\frac{\epsilon+i}{\epsilon}\right)-\Phi \left(1,1,-\frac{i}{\epsilon}\right)\right].$$
A numerical check, for $n=100$, shows this is very accurate:
Red curve is $w(2\pi)$ as a function of $\epsilon$, the blue dots are the smallest eigenvalue of the $100\times 100$ Toeplitz matrix.