You can fire any 2.5" shot shell. However, shotgun shells are not optimized for the Judge. So you won't necessarily get optimal results from it. Shotguns have long smooth barrels, usually minimum 18" long. The Judge, on the other hand, has a short rifled barrel -- sometimes as short as 2" (Public Defender) or as long as 6.5" (Raging Judge) but still, a short rifled barrel. So the net result is that you get really spread-out shot patterns. When using regular shotgun ammunition in a Judge, you can get sub-par performance -- because, frankly, the Judge isn't a shotgun, it's a short-barreled rifle, and general-purpose shotgun ammo was made for a big long shotgun with a smooth bore.
So -- will it fire? Yes, it will all fire, although some might cause you some problems when extracting the cases. For example, Nobel Sport 2.5" .400 buck in the blue box, that stuff jammed up my Judge every single time. The casings would expand too far, and keep the cylinder from turning for the next shot. So they would fire fine, but you'd then have to actually get out a knife and cut the extruding portion of the case so you could open up the cylinder, and then you'd need to use something like a pencil to force the expanded case out of the cylinder. So that's definitely something you don't want to be doing when it's important, y'know?
The rounds that work best for the Judge are the rounds that are made and marketed for it -- the Fed 410 handgun, the Hornady 410 Critical Defense, the NobelSport Judge 000 buck, the Winchester PDX1... those are all made for and optimized for the Judge, they give the best patterning, and the cases are made to be easily extracted from a revolver's cylinder.
Winchester birdshot will work, it'll fire, it'll just spread out really quickly, way faster than it would from a full-size shotgun. Don't worry about the velocity; velocity is a factor of the powder charge and the weight of the ammo, and a little tiny load like 1/2 ounce is going to put up very little inertial resistance, so the powder load will blast that out really fast.
Regarding slugs -- those are pretty much pointless with the Judge, as you could always just shoot .45 Colt rounds instead, and the .45 Colts are much more powerful because they're a lot heavier (up to 250 grains, vs. a normal 410 slug's 1/5 ounce or about 90 grains. A 250-grain lead bullet or 230-grain JHP is going to be a far more destructive projectile than a little 410 slug, and the accuracy from the 410 slug is not going to be anywhere near as good because the 410 slug is not very effective at engaging the rifling in the Judge's barrel. The 410 slug is 0.41" in diameter, but the barrel on the Judge is made for a 0.452" bullet. The bigger bullet of the 45 colt will engage the rifling and spin, becoming more stable and flying a lot more accurately. The slug, on the other hand, may fly erratically, and may even "keyhole" (meaning that it's not traveling true and straight like a spinning football, but that it's tumbling out of control; that's terrible for accuracy).
. I found some Federal 000 ammo and excited to try it, stopped in at a Walmart after, curiosity drove me to their ammo section and i found Winchester 410 shotgun ammo 2.5", 1275 velocity, 1/2 oz. 6 shot, there was also a version of 410 slugs. None of these said "hand gun". Are they safe? The velocity seems high, is this a "no, no" like +P 45's? Are slugs safe? or not worth it and stick to 45's? 
