Supporting EINK on mine could potentially help make it more visually compelling like yours, but I've done other EINK display projects in the past (see https://github.com/Mrjohns42/DoggieClock) and screen burn-in was definitely an issue.
Burn-in is very possible. It is ink particles inside a solvent. Drive it the wrong way and you'll damage the ink or the solvent. Just like LCD where if you drive it the wrong way, the liquid crystal gets damaged. The difference is LCD voltage waveforms are built into hardware and are much simpler since there's no moving components.
I find this very surprising, I was aware that you can sometimes get ghosting-effects, but I did not think permanent burn-in was possible on EINK tech.