EDIT: I misread the question and proved something easier. Oh well.
Any monic polynomial $p(x)=p_0+p_1x+p_2x^2+...+x^n$ with coefficients in a ring $R$ is the characteristic polynomial of a matrix with coefficients in $R$. Consider a vector space with basis $e_0,\ldots,e_{n-1}$, and the linear transformation that sends $e_i\mapsto e_{i+1}$ and $e_{n-1} \mapsto p_0e_0+p_1e_1+\cdots$
This linear transformation obviously has minimal polynomial $p(x)$, and so that must be the characteristic polynomial.
Any of the usual bases of symmetric functions is integer if and only if any other is, so we are done.
