Roughly 2% of white dwarfs harbor planetary debris disks detectable via infrared excesses, but only a few percent of these disks show a gaseous component, distinguished by their double-peaked emission at the near-infrared calcium triplet. Previous studies found most debris disks around white dwarfs are variable at 3.4 and 4.5 \(\mu\)m, but they analyzed only a few of the now 21 published disks showing calcium emission. To test if most published calcium emission disks exhibit large-amplitude stochastic variability in the near-infrared, we use light curves generated from the unWISE images at 3.4 \(\mu\)m that are corrected for proper motion to characterize the near-infrared variability of these disks against samples of disks without calcium emission, highly variable cataclysmic variables, and 3215 isolated white dwarfs. We find most calcium emission disks are extremely variable: 6/11 with sufficient signal-to-noise show high-amplitude variability in their 3.4-\(\mu\)m light curves. These results lend further credence to the notion that disks showing gaseous debris in emission are the most collisionally active. Under the assumption that 3.4-\(\mu\)m variability is characteristic of white dwarfs with dusty debris disks, we generate a catalog of 104 high-confidence near-infrared variable white dwarfs, 84 of which are published as variable for the first time. We do near-infrared spectroscopic follow-up of seven new candidate 3.4-\(\mu\)m variables, confirming at least one new remnant planetary system, and posit that empirical near-infrared variability can be a discovery engine for debris disks showing gaseous emission.