$\newcommand{\Z}{\mathbf{Z}} \newcommand{\Mod}{\mathrm{Mod}}$Note that if $\Mod^{\geq 0}_\Z$ denotes the category of connective $\mathrm{H}\Z$-module spectra, then by the Dold-Kan correspondence and the Schwede-Shipley theorem, there are equivalences of categories
$$\Mod^{\geq 0}_\Z \simeq \mathrm{Ch}_{\geq 0}(\Z) \simeq \mathrm{Fun}(\Delta^{op},\mathrm{Ab}) = s\mathrm{Ab}.$$
(You could interpret "category" as either $\infty$-categories or model categories, but the latter means that "equivalences" has to be replaced with "zig-zag of Quillen equivalences". I'll just sweep this distinction under the rug.) Since simplicial abelian groups are Kan complexes, we're asking for a characterization of the image of $\Mod^{\geq 0}_\Z$ under the equivalence of categories between connective spectra and infinite loop spaces (which are grouplike $\mathbf{E}_\infty$-objects in spaces).

Here is one such characterization. Let's model grouplike infinite loop spaces $X$ as functors $X:\mathrm{Fin}_\ast\to \mathrm{Top}$ such that $\pi_0 \mathrm{Map_{Top}}(Y,X)$ is an abelian group for all spaces $Y$ (i.e., $X$ is grouplike) and such that the map $X([n])\to X([1])^n$ is an equivalence. Such an object should be in the image of $\Mod^{\geq 0}_\Z$ iff it is somehow "strictly commutative". One way to characterize this is as follows. Let $\Lambda$ denote the full subcategory of the category of abelian groups spanned by the groups $\Z^n$ with $n\geq 0$, so there is a functor $\mathrm{Fin}_\ast\to \Lambda$. Then an infinite loop space is in the image of $\Mod^{\geq 0}_\Z$ if and only if the functor $\mathrm{Fin}_\ast\to \mathrm{Top}$ classifying it factors through a finite-product-preserving functor $\Lambda \to \mathrm{Top}$. In other words, $\Mod^{\geq 0}_\Z$ is equivalent to the full subcategory spanned by the grouplike objects in the category $\mathrm{Fun^{prod}}(\Lambda, \mathrm{Top})$. This is a very strong condition to impose on an infinite loop space; for example, $\mathbf{C}P^\infty$ admits such a factorization, but $BU$ (with either the additive or multiplicative infinite loop space structure) doesn't.