**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,...,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.