Condition
$$\lim_{n\to\infty}\frac{\log|c_n|}{n}=-\infty$$
is sufficient for $f=0$.

Since $f(z)=e^{-cz}$ and $c_n=e^{-cn}$ satisfy all
conditions, we see that this is best possible in certain sense. 

This follows for example from a (much more general) theorem of N. Levinson, Gap and density theorems, AMS, 1940, page 121.
Levinson's theorem allows some growth of $F$, and much more general class of
sequences instead of integers.

Remark. In fact Levinson generalizes a theorem of Vladimir Bernstein 1932
(Theorem 32 in Levinson's book), which also implies the result that I stated.