Is there an analog of the Barratt-Eccles construction for group-like E_∞-spaces and E_∞-ring spaces? The Barratt-Eccles operad is an operad in simplicial sets
that provides a particularly nice model of an E∞-operad;
algebras in spaces over the Barratt-Eccles operad model E∞-spaces,
i.e., homotopy coherent commutative monoids in spaces.
It can be described concretely by applying the nerve functor
componentwise to an operad in groupoids, which itself is obtained by
applying the codiscrete groupoid functor componentwise
to an operad Σ in sets such that Σ(n) is the symmetric group of order n
and the operadic composition Σ(n)×(Σ(a₁)×⋯×Σ(aₙ))→Σ(a₁+⋯+aₙ)
is given by stacking the permutations in Σ(aᵢ) together and composing them
with the block permutation in Σ(a₁+⋯+aₙ) induced by the permutation in Σ(n).
Here the codiscrete groupoid functor sends
a set X to the groupoid with X as the set of objects
and exactly one morphism between any pair of objects;
it is the right adjoint to the forgetful functor
from groupoids to sets that sends a groupoid to its
underlying set of objects.
I am interested in similarly spirited constructions
for various cousins of E∞-spaces.
Specifically, I am interested in
group-like E∞-spaces, which can be thought of as homotopy coherent commutative groups and are a model for connective spectra.
Another interesting case is E∞-ring spaces,
which can be thought of as homotopy coherent commutative rings,
and are a model for connective E∞-ring spectra.
As pointed out by Peter May in his answer,
operads cannot model such structures
because they do not allow for operations
with multiple outputs, e.g., diagonal maps,
so a part of the question is what type of structure one should use.
For example, simplicial algebraic theories (see http://ncatlab.org/nlab/show/(%E2%88%9E,1)-algebraic+theory) seem to be a viable option.
In particular, the Barratt-Eccles construction
admits a particularly elegant formulation in terms of a
(2,1)-algebraic theory (i.e., a groupoid-valued algebraic theory),
see http://ncatlab.org/nlab/show/(2,1)-algebraic+theory+of+E-infinity+algebras.
Is there an analog of the Barratt-Eccles construction for group-like E∞-spaces and E∞-ring spaces?
 A: There are serious problems making your ideas coherent here!  The notion of operad
was in large part intended to model kinds of algebras whose laws do not involve repeated
variables, do not involve diagonal maps.  Operads do not even model groups, in particular,
and that was intended.   Any $E_{\infty}$ operad models all connective spectra, not just connected ones, the essential point being that the zeroth space of the spectrum associated to an $E_{\infty}$-space $X$, no matter how constructed, must be a group completion of $X$.
There is no "the" $E_{\infty}$ operad, rather there are many interesting ones.
(There is an axiomatization of infinite loop space machines, due to Thomason and myself,
that makes this precise.)   If you want to model $E_{\infty}$ ring spectra using operads
only, you can make a mistake by trying to use just one operad, as I did over 40 years ago, 
or you can do it right by using two interrelated operads, one for the addition and one for
the multiplication, as I also did over 40 years ago. See http://www.math.uchicago.edu/~may/PAPERS/Final1.pdf
for a modern recapitulation of that early theory.
