= Solution
Take a <Hankel contour> $C_\eta$ around the positive real axis, going from infinity on its lower side toward a circle of radius $0<\eta<1$, clockwise around that circle, then out along the upper side. Use $0\le\arg z\le2\pi$ and define
$$
H(t)=\int_{C_\eta}\frac{z^te^{-z}}{1+z}\,dz.
$$
This is an <entire function> of $t$: the small circle stays away from zero and the pole at $-1$, while the exponentially decaying rays converge uniformly for $t$ in <compact subsets>. <Contour deformation> shows independence of $\eta$. For $\Re t>-1$, let $\eta\to0$. The upper ray contributes $F(t)$, the lower ray contributes $-e^{2\pi it}F(t)$, and the small-circle contribution vanishes. Thus the <analytic continuation> is
$$
\boxed{F(t)=\frac{H(t)}{1-e^{2\pi it}}.}
$$
The only apparent singularities are at integers. At the <nonnegative integers> they are removable because the original integral is <holomorphic> throughout $\Re t>-1$. Hence singularities can occur only at $-1,-2,\ldots$.
In fact all of those points are <simple poles>. At $t=-n-1$ the two rays cancel and the clockwise circle gives $H(-n-1)=-2\pi i\,c_n$, where
$$
c_n=[z^n]\frac{e^{-z}}{1+z}=(-1)^n\sum_{j=0}^n\frac1{j!}\ne0.
$$
Since the denominator has derivative $-2\pi i$ at each integer, the residue at $-n-1$ is $c_n$. The parameter has been called $t$ consistently; the final use of $z$ for this parameter in the printed question is harmless notation.
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