Let's take the strictest meaning of "similar evaluation", i.e., a product formula. The by the remark, the answer is probably no, because for instance the Stirling number s(9,3) = 118124 = 2*2*29531.
A different meaning is whether there is some explicit combination of nested sums and products for a Schur evaluation at consecutive integers. I don't know the answer to that. Unlike for a straight product formula, it is difficult to look for such formulas or rule them out.
The loosest possible meaning is whether there is an efficient way to evaluate sλ(1, 2,...,n). The answer to that is an easy yes, because you can use the Jacobi-Trudi determinant.