Sorry this got too long for a comment, so I post it as an answer. 

A CW-complex equipped with a $G$- action is not the right thing to consider.  A $G$-CW complex is a space, that can be built using blocks of the form $G/H \times (D^n,S^{n-1})$ (in the same way you build a CW-complex). A $G$-CW complex has better properties than a CW-complex, for example the fixed point set of any subgroup is a subcomplex. However most CW-complexes with a $G$ action can also be given the structure of a CW-complex. 

In the example of $G=\{1,t\}=\mathbb{Z}/2$ acting on $S^1\times S^1$ by flipping the components, one could for example take the following $G$-CW- structure with 1 $0$-cell $P$ of type $G/G$, two 1-cells $A,B$ of type $G/1,G/G$ and 1 2-cell $C$ of type $G/1$:

![image of G-CW structure ][1]

Then the cellular chain complex may be considered as $\mathbb{Z}[G]$ chain complex. Here it is:

$\mathbb{Z}[P]\leftarrow\mathbb{Z}[A,tA,B] \leftarrow \mathbb{Z}[C]$

, where the differentials are given by $A,B\mapsto 0, C\mapsto A+tA+B$. Forgetting the names, we can write the chain complex as 

$\mathbb{Z}[G/G]\leftarrow\mathbb{Z}[G/1]\oplus \mathbb{Z}[G/G] \leftarrow \mathbb{Z}[G/1]$

These maps are $G$-equivariant. If one would apply $\otimes_{\mathbb{Z}[G]}\mathbb{Z}$ to this chain complex and take the homology, one would just get the cellular homology of the quotient $H_*(X/G)$. 

I think, that if one takes a projective resolution of this chain complex first, one should get $H_*(X\times_GEG)$. (The cellular chain complex of $X\times EG$ is a free resolution of the cellular chain complex of $X$). 

  [1]: http://www.freeimagehosting.net/uploads/d0acae4eb9.jpg