What would make diareeha smell of burnt matchsticks?

Hydrogen sulfide: really good description. That smell is sulfur from bacterial breakdown of foods that have sulfur You cannot create sulfur out of nothing anymore than you can create gold, it's got to come from something you ate (eggs, most commonly). Only some bacteria create it. They don't have to be disease causing, although helicobacter (stomach germ) both creates sulfur smell AND irritate stomach.

Infectious diarrhea: Diarrhea from invasive organisms or those that cause bleeding like IBD may produce such smell