I take an eternalist to be someone who believes there are no past/future things, it all exists "at once". I understand the position as it applies to concrete objects, but I am not so clear on what the position says about past/future events and properties of past-ness/futurity as such.
For instance, some eternalists appears to say that there is no absolute past event like "I slept at 10 yesterday." But there is a presently past event like "It is the case that I slept at 10 yesterday" (in fact, they even appeal to events like these to account for causations, which they take to be synchronic). But what makes it the case that "I slept at 10 yesterday" is false, whereas "it is the case that I slept at 10 yesterday" is true?
A similar worry applies to properties of pastness and futurity as well. On the one hand, an eternalist wants to say these properties are uninstantiated by concrete objects. On the other hand, they do not want to deny those properties as fictional themselves (thus, they are not like say "square circularity" or "magical"). But are they then admitting uninstantiated universals into their ontologies?