What do the "old man" and the "sweet boy" symbolize to the speaker?

For Whitman, the universality of the death experience is an intrinsic part of the war experience.  Whitman sees no other resounding truth other than the dead that results from war.  While the Civil War might have been fought for justifiable reasons, Whitman makes it clear that those reasons become secondary in the mounting death count that is such a part of war.  The old man that is dead represents how war ravages the old of any society.  Whitman makes clear that the elderly in any society are ruined by war.  For Whitman, the elderly in a social setting should be the source of life, the element of experience, and wisdom.  In his description of the elderly soldier, Whitman sees something far from this:

so gaunt and grim, with well-gray'd hair,
and flesh all sunken about the eyes.

This vision of the elderly being reduced to "gaunt and grim" and "flesh all sunken about the eyes," shows how the old in war, the elderly in a social setting, are easily eliminated.

For Whitman, the child he sees represents the young.  Whitman clearly understands that one of the most damaging elements of war is how the youngest members of society are wiped out as a result of it.  When Whitman describes the "sweet boy," he suggests a picture of youth cut down in its prime.  With "cheeks yet blooming," Whitman makes clear that while the old dying is a painful reality, it is matched with the death of the young.  Conveying the symbolic notion of hope being removed is where Whitman seeks to convey to the reader the true and horrific nature of the war experience.