Several Reformed theologians hold to the position that the "person" of the Son is generated whereas the Son's participation in the "divine essence" is not.
For the strictest of Clarkians this would be rather bizarre, since for them the person of the Son is the sum of His predicates. Tautologically speaking, the person of the Son is the definition of the Son, and "definition" is precisely what Clark equated with "essence." In other words, if the person of the Son is generated, that would seem to imply, given Clark's terminology, that the essence (definition) of the Son is generated.
This would be a problem if they wish to hold, as did Clark, that the Son is autotheos. But I suspect this is, in fact, probably not a case of inconsistency so much as an inconsistent reliance on Clark's definitions. I imagine that these Clarkians who argue that the Son's essence is not generated mean that the divine nature or set of attributes of the Son is not generated. In other words, the Father doesn't "communicate" the [or a] divine nature to the Son if such implies the divinity of the Son is in some sense derived from the Father; rather, Clark (and probably these Clarkians) thinks "communicate" merely suggests that the Son has the [or a] divine nature "in common" with the Father. This would suggest that the Son may well communicate the [or a] the divine nature to the Father, but the strange sound of this suggestion is not the focus of this post.
The main point of this post is what I now turn to: even if one holds to this now qualified Reformed view that 1) the Son is autotheos, and 2) the person of the Son is generated but not His essence, terminological difficulties remain in respect to the meaning of "generation." If the Son is autotheos, how is it that His "person" can be generated? What does that mean? Generation is a word which implies logical, if not temporal, derivation. But if the "second person" of the Trinity is self-existent and autotheos, He cannot be said to be eternally generated; that is, the second person wasn't "generated" per se, since the second person qua second person is allegedly underived, self-existent, and God-of-Himself.
Rather, it would appear that in this case, "eternal generation" actually means the second person merely eternally assumed the personal property of Sonship. If the second person of the Trinity only derives His property of Sonship from the Father, it's misleading to say the second person qua second person is eternally generated. For the person of the Son isn't generated at all.
At most, the second person derives His personal property from the Father, not His person from the Father. But then it would be more appropriate to refer to the Son as eternally adopted. Furthermore, this neo-Adoptionism appears to make the relationships among the persons arbitrary, for I see no reason to think that the personal properties could not be reversed. To us, the first person may be the Father and the second may be the Son, but why may it not have been the case that the first person be the Son and the second the Father? Or, why must the second person have "derived" His property from the first if each of the persons of the Trinity are autotheos, ontologically co-equal in every respect?
I cannot see how eternal generation is compatible with the view that the persons of the Trinity are each autotheos and self-existent.