The poem explains how everything has gone wrong for poor old
Bill the Bullocky finally he gets bogged 9 miles from Gundagai,
then the dog shat in the tuckerbox. Probably the only dry spot near
the flooded creek.
Old Answer:
The meaning of 'the dog shat in the tuckerbox" is that the dog
(a four legged domesticated animal descended from the wolf)
defecated into (as opposed to beside or underneath) a container of
food (consumable items).
Phrases carrying a similar meaning would be things such as:
'Lassie did number twos inside the lunchbox' OR 'The poodle had
a bowel movement within the picnic basket.'