(Edited to reflect adjusted question.) The topology of symmetric products of a non-singular curve was worked out by Macdonald, Topology 1 (1962), 319-343. The starting point is the observation (which is quite general, hence applies to singular curves) that the rational cohomology of the symmetric product is the invariant part of the cohomology of the Cartesian product. From that you , from which one can in principle work out the Euler characteristic, though I don't know whether you'll be able to find as compact a formula as the one gave in the smooth case.
You ask also about the
The Euler characteristic of the Hilbert scheme of $n$ points on a curve . This is a rather subtler problem, since these spaces feel the nature of the singularities. In the simplest case, a curve with one node, I wrote down how to describe the Hilbert scheme as a variety with normal crossings here (see Section 3.2 and the appendix). I also described its cohomology. From that you can write down the Euler characteristic.
