Meyer's argument begins with premises that are hilariously absurd. Defining entities as being able to be causes of themselves? Having "entities" even able to be "causes"?

I think this is mostly a presentational issue. The purpose of the argument was to construct a non-strict partial order "<=" out of the causal relation, and that requires x<=x. This is just to enable the application of Zorn's Lemma.

To avoid the hilarity of things being causes of themselves, we could easily adjust the definition of <= so that "x<=y" if and only if "x=y or x is a cause of y". Or the argument could be presented using a strict partial order <, under which nothing will be a cause of itself. The argument doesn't need to analyse "entity" or "exists" since such an analysis is inessential to the premises.

And finally, please remember that the whole thing was not meant to be taken seriously; though rather amusingly, Alexander Pruss (whose site I linked to) apparently has been treating it as a serious argument. Oh dear.