There is a paper by Brion: "Poincaré duality and equivariant (co)homology," *Michigan Math. J.* 48 (2000).  As mentioned in one of the comments, he uses Borel-Moore as the homology theory.

That said, one should always be careful about the word "duality" in this context -- often people just mean there's an *isomorphism* $H_G^k X \rightarrow H^G_{n-k}X$.  (Which does exist using equivariant Borel-Moore homology, when $X$ is smooth.)  The geometric meaning is the same as in the ordinary case: this isomorphism can be used to define the "class" of a codimension $k$ $G$-invariant sub(manifold/variety/cycle/etc) $[V]$ in $H_G^kX$.

Duality could also mean a perfect pairing $H_G^* X \otimes H_G^*X \rightarrow H_G^*(point)$, given by the equivariant pushforward (integral), and coming from this, dual bases for $H_G^*X$ as $H_G^*(point)$-modules.  Again, the geometric meaning is similar to the ordinary case: if you're lucky enough to have invariant subvarieties whose classes form a basis for $H_G^*X$, the Poincaré dual basis is given by classes of subvarieties intersecting the original ones transversally (if they exist).  However, the existence of dual bases is a stronger statement, because the equivariant integral is generally nonzero on classes of degree greater than $\dim X$.

(I should say all this is prejudiced toward the equivariant cohomology rings that usually show up in algebraic geometry, e.g., $H_G^*X$ is a free over $H_G^*(point)$, so that it makes sense to talk about dual bases.)