The discovery paper (Su et al. 2010) mentions that bipolar shells on the order of $0.5\text{ kpc}$ were found on both sides of the nucleus of Centaurus A (Quillen et al. 2006). Centaurus A is a galaxy of contested morphology, likely either elliptical or lenticular - certainly not a spiral. This would seem to be a strong indicator that similar bubbles can indeed be found in non-spiral galaxies.
Now, the nuclear shells Quillen et al. detected were seen at mid-infrared wavelengths (5.8, 8.0 and 24 $\mu\text{m}$), rather than via the gamma rays with which Su et al. discovered the Fermi bubbles. However, the authors believe they may have been caused by a burst of star formation a few million years ago, driven by winds. This is one of two main scenarios considered by Su et al. for the formation of the Milky Way's bubbles (the other being AGN activity), meaning that the same processes could indeed have been at work in both galaxies. I don't know if corresponding gamma-ray emission from the nuclear bubbles in Centaurus A has since been detected, but it certainly seems plausible.
I find it a little strange that Pshirkov et al. were so careful to focus on spiral galaxies. It's clear that starburst galaxies and the host galaxies of AGN can obviously have non-spiral morphologies. Perhaps the group was just interested in searching for other nuclear bubbles on galaxies similar to the Milky Way which might have similar conditions and evolutionary histories.