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,...,e_{n-1}, and the linear transformation that sends e_i->e_{i+1} and e_{n-1} -> p_0e_0+p_1e_1+...
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.