People tried to make a "Microgun",which was a man-portable minigun. It was deemed unfeasible,since the ammo used lacked the stopping power required. They had to downsize the bullets to keep the culminative recoil from knocking the operator on their ass. Standard miniguns,which are mounted to walls,vehicles,and aircraft,use either 7.62mm,5.56mm,or .50 cal ammo. It fires RIFLE ROUNDS. Which have quite a bit of kick.
Now,a twelve gauge round? Well,those possess considerable kick,more than rifle rounds,I think. While,yes,a minigun spraying 12 ga. buck or 12 ga. slug at 10,000 RPM would be impressive,and cool,it would not be man-portable.
Note that miniguns were NEVER designed for personal use. It was designed to be a crew served weapon,as an evolution to the machine gun nest. Same bullet,higher rate of fire,therefore,more effective at supressing an advancing force. A minigun is,almost literally,six C9's or M60's strapped together,firing in series. The bigger ones are used for ANTI ARMOUR DETAIL. The "Warthog" jet? Has a GAU-8(I think...) mounted to the nose. It pops tanks open like a teenager pops zits. And,unless the Warthog's in a stoop (Steep,high speed dive),the recoil from this recoil-gas operated monster will STALL OUT THE JET.
Think about that. Also,miniguns tend to waste ammo. At 10k RPM,you better be firing at a huge crowd,or taking down a VERY real threat. Like an Exhusett* missile coming at a ship.
*:Sorry if I butchered the name. It's a French anti-ship missile that usually flies at Mach 0.9,and is reputed to have a 1:1 fired:sink ratio.