To add to T. Amdeberhan's answer, the inequality
$$|c_{n}|\leq1/n$$
is actually sharp since $nz+z^{n}$ is univalent in the open unit disk. Indeed, for $z\neq u$,
$$nz+z^{n}=nu+u^{n}\quad\Longleftrightarrow\quad -n=z^{n-1}+z^{n-2}u+\cdots+u^{n-1},$$
and the sum on the right-hand side has modulus less than $n$ for $z$ and $u$ in the open unit disk.

Moreover, concerning the other coefficients, the following result is proved in 

> Suffridge, T. J.
On univalent polynomials. 
J. London Math. Soc. 44, 1969, 496-504. 

Let
$$C_{k,j}=\frac{n-k+1}{n}\frac{\sin jk\pi/(n+1)}{\sin j\pi/(n+1)},\qquad k=2,\ldots,n,\quad j=1,2.$$
Then, for univalent polynomials **with real coefficients and such that $c_{n}=1/n$**, the coefficients $c_{k}$ satisfy the sharp inequalities
$$|c_{k}|\leq |C_{k,1}|,
$$
and for univalent polynomials **with real coefficients and such that $c_{n}=-1/n$**, 
$$
|c_{k}|\leq\begin{cases}
|C_{k,1}|\quad(n\text{ even})\\
|C_{k,2}|\quad(n\text{ odd}).
\end{cases}
$$