On that same field trip today, Alonso reported seeing a Wood Duck (Aix sponsa) carrying an egg. The bird was at some distance to him, and I didn’t see the bird. We discussed the sighting, and concluded that it was possible, but somewhat unusual.
There is a bit in the literature: Robert W. Strader et al. (The Wilson Bulletin 90:1, March 1978, pp. 131-132) report an observation and suggest that the purpose of egg carrying is to remove eggs damaged by a woodpecker, another animal, or mishap. Larry J. Hindman (The Wilson Journal of Ornithology 127:4, December 2015, pp. 759-761) reports an observation of a male Wood Duck carrying an egg, and discusses some other possible explanations for the behavior.
The removal behavior might be more common than we think. In our boxes, it’s sometimes been the case that we count x eggs on a given Sunday, with high certainty, and we count x-1 eggs the next Sunday.