From a Catholic perspective

The term "eternity" (and the accompanying adjective "eternal") is used several times in the Catechism of the Catholic Church, but is never defined there. In the Summa Theologica of Thomas Aquinas, there is a whole "Question" ("investigation" might be a better translation of the word) regarding eternity and God. Aquinas begins this investigation—Question 18 of the First Part of the Summa Theologica—by defining "eternity", following Boethius, as

the simultaneously-whole and perfect possession of interminable life.

He explains this definition:

As we attain to the knowledge of simple things by way of compound things, so must we reach to the knowledge of eternity by means of time, which is nothing but the numbering of movement by "before" and "after." For since succession occurs in every movement, and one part comes after another, the fact that we reckon before and after in movement, makes us apprehend time, which is nothing else but the measure of before and after in movement. Now in a thing bereft of movement, which is always the same, there is no before or after. As therefore the idea of time consists in the numbering of before and after in movement; so likewise in the apprehension of the uniformity of what is outside of movement, consists the idea of eternity.

Further, those things are said to be measured by time which have a beginning and an end in time, because in everything which is moved there is a beginning, and there is an end. But as whatever is wholly immutable can have no succession, so it has no beginning, and no end.

Thus eternity is known from two sources: first, because what is eternal is interminable---that is, has no beginning nor end (that is, no term either way); secondly, because eternity has no succession, being simultaneously whole.

[Note: "movement", as Aquinas uses it here, does not mean "change of position" but—as medieval philosophers often used the term—any sort of change, internal or external. God, according to this conception, is not capable of or susceptible to change.]

In other words, "eternity" is an experience of all that is, without any reference to "time", "before" or "after", "beginning" or "end", and without division into parts in any way.

Though he is certainly not a mainstream Christian theologian, Emanuel Swedenborg (1688-1772) does offer some statements on God's eternity, and on Jesus Christ in relation to God and eternity, that may be helpful in answering the question.

These are taken from his work True Christianity, originally published in Latin in Amsterdam, 1771. The first two are about the eternity of God itself. The third and fourth relate God's eternity to Jesus Christ--and are thus perhaps relevant to the basis of "the claim of Jesus Christ" mentioned in the question.

First quote:

God is infinite because he existed before the world, before space and time came into being. The physical world has time and space. The spiritual world, on the other hand, lacks actual time and space, although it does have apparent time and space.

Time and space were introduced into both worlds for the sake of distinguishing one thing from another, large from small, many from few - one quantity from another, and one quality from another. Time and space allow our bodily senses to discern the objects they are sensing; and they allow our mental senses to discern the objects they are sensing - to be affected, to think, and to choose. (True Christianity #29)

Second quote:

Since the world was made, God has existed in space independently of space, and in time independently of time. . . .

God is present in space independently of space and in time independently of time because God is always the same, from eternity to eternity. What God was like before creation, God was like after creation. Before creation, there was no space or time in and with God. After creation, there was. Because God remained the same, then, he is in space independently of space, and in time independently of time. As a result, nature is separate from him and yet he is omnipresent in it. (True Christianity #30)

Third quote:

Because God is present in all time independently of time, his Word [i.e., the Bible] speaks of past and future in the present tense. For example, in Isaiah: "A Child is born to us, a Son is given to us, whose name is Hero, Prince of Peace" (Isaiah 9:6). In David as well, "I will announce this decision: Jehovah said to me, You are my Son. Today I fathered you" (Psalms 2:7). These statements refer to the Lord who was to come. In the same source it also says, "In your eyes, a thousand years are like yesterday" (Psalms 90:4).

From many other passages in the Word about seeing and being vigilant we can see that God is present everywhere in the entire world, and yet there is nothing belonging to the world in him, that is, nothing limited in space and time. For example, this passage in Jeremiah:

Am I not a God near you, rather than a God far away? Can a man be covered over in hiding places so that I would not see him? I fill the whole heaven and the whole earth. (Jeremiah 23:23-24)

(True Christianity #30)

Fourth quote:

The being who came into the world was God himself, who from eternity [has been and] is the One. This is very clear from the birth of the Lord and Savior. He was conceived by the power of the Highest through the Holy Spirit. As a result the Virgin Mary gave birth to his human manifestation. It follows then that his soul was the Divinity itself that is called the Father--God is, after all, indivisible--and the human being born as a result is the human manifestation of God the Father, which is called the Son of God (Luke 1:32, 34, 35). It follows from all this that when we turn to the Lord God the Savior, we are turning to God the Father as well. This is why he replied to Philip, when Philip asked him to show them the Father, "Those who see me see the Father. How then can you say, 'Show us the Father'? Do you not believe that I am in the Father and the Father is in me? Believe me that I am in the Father and the Father is in me" (John 14:6-11). (True Christianity #538)

So according to Swedenborg's theology, God's eternity is based on God's being above and outside of time, time being a property that God created as part of the material universe. The spiritual universe, he says, does not have time, but does have an analog of time such that those living there experience a passage of events. God, however, is present in all time simultaneously from a state that exists outside of time. This also means that God is unchangeable and unchanging, and therefore eternally one and eternally the same being.

In Swedenborg's theology, Jesus Christ is God himself entering from eternity into time and space. If that doesn't wrap your brain in knots, I don't know what will! But the whole concept of God from eternity entering into the world of time and space is a huge challenge to our time- and space-bound brains.

Perhaps the most understandable way to express it is that Jesus Christ is the human manifestation of the eternal, omnipresent God in the material, time- and space-bound world that we humans inhabit.

Even the Incarnation of God in time as Jesus Christ did not change God--as it would appear to our time-bound minds. God is eternally present in all time and space. Therefore God's Incarnation as Jesus Christ is always a present reality for God, just as everything else that we perceive as past, present, and future is a present reality for God--who experiences it all simultaneously from timeless, spaceless infinite state of Divine Being.

Therefore, from the perspective of Swedenborg's theology, in terms of the question, "the claim of Christ" would be based on "an eternal moral order" rather than on "an individual interest in a parochial conflict that first arose at a historical time."

Having said that, since God did enter human history as Jesus Christ at a particular point in material-world space and time--meaning at a particular time in history and in a particular geographical location inhabited by a particular set of cultures--his life, ministry, and teaching necessarily took a particular form based on that human culture. We therefore must use our thinking minds to sort out the eternal divine truth and love in his life and teachings from the time- and space-bound forms in which that eternal divine truth and love were expressed.

But that is a vast, complex subject that can in no way be done justice in this compact question and answer format.

This may perhaps offer an answer to your question based on the theology of Emanuel Swedenborg as understood within the denominations that were subsequently founded based on his theology.