Is there an analogue of CW-complexes built from $K(\mathbb Z, n)$ instead of $S^n$?

The answer is yes. You mention model categories, so I guess what you have in mind is right Bousfield localization. Given a nice enough model category $M$ (spaces or simplicial sets qualify), and a set of objects $K$, the right Bousfield localization $R_K(M)$ is new model structure where the objects of $K$ are the new cells, and a map $f:X\to Y$ is a weak equivalence (a $K$-colocal equivalence) if for every $A\in K$, the map of simplicial sets $map(A,X)\to map(A,Y)$ is a weak equivalence. These are the maps seen to be weak equivalences by $K$.

This model structure has the same fibrations as $M$. The cofibrant objects are precisely the $K$-colocal objects, i.e. objects $W$ such that $map(W,X)\to map(W,Y)$ is a weak equivalence for all $K$-colocal equivalences $W$. If every object of $M$ is fibrant (e.g. Top) then $R_K(M)$ is cofibrantly generated, with the same generating trivial cofibrations $J$ as $M$, and with generating cofibrations defined as $\overline{\Lambda(K)} = J \cup \{A^*\otimes \partial \Delta[n]\to A^*\otimes \Delta[n] | A\in K, n\geq 0\}$, where $A^*$ is a cosimplicial resolution of $A$. Morally, this is saying that the objects of $K$ are the new cells, and cofibrant replacement in this new model structure entails building things out of the objects in $K$.

In your case, taking $M = Top$ and $K = \{K(\mathbb{Z},n)\}$ will accomplish what you asked for. All the facts I wrote above are in Hirschhorn's book. A source for lots of examples is my paper (with Donald Yau): Right Bousfield Localization and Operadic Algebras

Repeating my comment above as answer:

You may find Peter May's article "The Dual Whitehead Theorems" interesting in this context. It's at math.uchicago.edu/~may/PAPERS/47.pdf