Not able to find an incendiary shotgun? Here is a quick way to get searching for one.
Start a new game of Breakdown. Since Breakdown allows you to anywhere immediately, we're going to loot the two gun shops in the game. Grab the first vehicle you see and go to the two locations indicated in the maps below (one in Spencer's Mill, one in Marshall).
Once at a gun shop, search the entire building for the incendiary shotgun. If its not in either building, just restart Breakdown and search these buildings again. Rinse and repeat until shotgun found.
The only negative of this is that once you find the shotgun, you may not have enough ammo to kill 25 zeds. You may luck out and find the ammo in the gun shop, or if you're really skilled you can take out multiple zeds in one shot. If you're unlucky/unskilled like me however, you'll now need to construct a Munitions Workshop at a home base (Establish base -> Basic Workshop -> Workshop -> Munitions Workshop). The second to last option in the Munitions Workshop menu is incendiary shotgun shells. Setting this up will take time, but once you do, you can create as many shells as needed to complete the achievement.