There is a brand of shoe called "Hush Puppy" and it's is named that as a reference to the expression, "My dogs are barking." In the case of the shoe, Hush Puppies are supposedly so comfortable they can quiet your puppies - that is, stop your feet from hurting.
I'm uncertain of its origin. I started hearing this phrase around 2002 or so from different people, then it was used by one of the soldiers in "Half Life: Opposing Force"
I originally took this to mean "I gotta take a DUMP!" But have been proven wrong , apparently. Nonetheless I still prefer my interpretation.