Kan's category of semisimplicial spectra and the category of sequential spectra of pointed simplicial sets together with the Kan suspension are models for spectra with Eilenberg-MacLane spectra of the desired form.

Ken Brown equipped the category of semisimplicial spectra with a model structure in which every object is cofibrant and group objects are fibrant. Bousfield and Friedlander showed that there is a zig-zag of Quillen equivalences between the model categories of semisimplicial spectra and of Bousfield-Friedlander spectra with the stable model structure. In particular, the homotopy category of semisimplicial spectra is the stable homotopy category.

Kan's stable Dold-Kan correspondence asserts that the category of abelian group objects of semisimplicial spectra is equivalent to the category of unbounded chain complexes of abelian groups. Considering an abelian group $A$ as an unbounded chain complex concentrated in degree $0$ yields under this correspondence an abelian group object $HA$ in semisimplicial spectra that models the Eilenberg-MacLane spectrum and that is both cofibrant and fibrant.

The zig-zag of Quillen equivalences by Bousfield and Friedlander passes through the category of sequential spectra of pointed simplicial sets with the Kan suspension. More precisely, there is a right Quillen equivalence $\mathrm{Ps}$ from the category of semisimplicial spectra to the category of sequential spectra with the stable model structure. For any semisimplicial spectrum $X$, the structure maps of $\mathrm{Ps}(X)$ are monomorphism. Thus $\mathrm{Ps}(X)$ is cofibrant. Hence if $X$ is a group object (and therefore in particular fibrant), then $\mathrm{Ps}(X)$ is cofibrant, fibrant and a group object as well, since $\mathrm{Ps}$ is a right Quillen functor. For the semisimplicial spectrum $HA$ above, the sequential spectrum $\mathrm{Ps}(HA)$ is an Eilenberg-MacLane spectrum of the desired form and is explicitly given by the sequence of pointed simplicial sets 
$$A, \overline{W}A, \overline{W}\overline{W}A, \ldots, \overline{W}^n A,\ldots$$
where $A$ is considered as a constant simplicial abelian group and $\overline{W}$ is the right adjoint of Kan's loop group functor.