I believe 0.110.x mishandles badges with an entity-filter.
I have four badges whose visibility is controlled by an entity-filter and in 0.110 none are displayed.
If I open the Raw Configuration Editor, close it, exit UI editing mode, all entity-filter controlled badges are displayed (even if some should be hidden). Refreshing the page causes all of them to disappear (even though some should continue to be displayed).
My Lovelace configuration has not changed, I don’t recall reading any breaking changes for entity-filter, so this must be a bug. I’ll be reporting it in the front-end GitHub repository.
NOTE: I am using 0.110.3 and do not have the problem you described regarding disappearing quotes.
I have several entity-filters for badges and only the one that use '==' loses its quotes upon saving the file. The other ones that use '!=' or 'on' don’t lose their quotes. I don’t know if this old or new behavior but it seems odd to me.