While only the female incubates and broods, both parents participate in feeding. The parents of these chicks have been busily flying to and from the box. On most visits, a parent enters the box with a food item, feeds it to a chick and leaves. Occasionally, the parent will exit the box carrying something.
The white thing in the wren's bill is a fecal sac. The feces of young nestlings are encased in a mucous membrane. This makes it easy for parents to keep the nest clean by removing feces. A parent will usually carry and drop the fecal sacs at some distance from the nest to keep the nest's location less obvious to predators.
House Wren chicks usually stay in the nest for a little more than two weeks before fledgling. If that holds true for this group, they should fledge during the first week of July.