Origen does seem to have included demons in his concept of apocatastasis, but this specific teaching as well as the doctrine of apocatastasis in general - the teaching that everyone will be saved in the end - was roundly condemned by the local Council of Constantinople in 453. The acts of this Council were later accepted by the whole Church at the 5th Ecumenical Council in 553, which explicitly excommunicated anyone following his teachings. Canon XV of this Council reads:
If anyone shall say that the life of the spirits (νοῶν) shall be like
to the life which was in the beginning while as yet the spirits had
not come down or fallen, so that the end and the beginning shall be
alike, and that the end shall be the true measure of the beginning:
let him be anathema.1
All told, there are 24 anathemas against the teachings of Origen that came out of this particular council.
The teaching is summarized in Michael Pomazanski's Orthodox Dogmatic Theology (3rd. ed):
The teaching of a restoration (apokatastasis) of creation at the
Second Coming of Christ is found in Acts 3: 19– 21: Repent ye,
therefore, and be converted, that your sins may be blotted out, when
the times of refreshing shall come fom the presence of the Lord; and
He shall send Jesus Christ, Who was before preached unto you: Whom the
heaven must receive until the times of restitution (apokatastaseos)
of all things*.... Here apokatastasis is to be understood as the
change, renewal and transfiguration of man and the cosmos at the time
of the General Resurrection, as outlined in the present chapter. It is
not to be understood as “universal salvation,” that is, the heretical
Origenist notion that all human beings and even all the demons will
ultimately enter into everlasting blessedness. The Orthodox
understanding of apokatastasis is presented in the Ambigua of St.
Maximus the Confessor, which contain both a refutation and a
correction of Origenism. (See St. Maximus the Confessor, On the
Cosmic Mystery of Jesus Christ, p. 56.)2
A primary text of Origen's in this regard seems to come from Book III of his First Principles, wherein it is written:
Into this condition, then, we are to suppose that all this bodily
substance of ours will be brought, when all things shall be
re-established in a state of unity, and when God shall be all in all.
And this result must be understood as being brought about, not
suddenly, but slowly and gradually, seeing that the process of
amendment and correction will take place imperceptibly in the
individual instances during the lapse of countless and unmeasured
ages, some outstripping others, and tending by a swifter course
towards perfection, while others again follow close at hand, and some
again a long way behind; and thus, through the numerous and uncounted
orders of progressive beings who are being reconciled to God from a
state of enmity3
Some of the writings of Gregory of Nyssa also supported Origen's concept of apocatastasis, which he himself appeared to have accepted from Origen. These writings were subsequently rejected, however, by numerous Church Fathers, including Barsanufius the Great, Herman of Constantinople, Mark of Ephesus, Maximus the Confessor, and Photius the Great.
Origen was the pupil of Clement of Alexandria. Clement was charged later with heresy by Photius the Great for some of his Christological teachings, but I could not find any evidence that he influenced Origen's doctrine of apocatastasis (Clement of Alexandria is not acknowledged as a Church Father by the Eastern Orthodox Church. He is not to be confused with Clement of Rome or Cyril of Alexandria, both of whom are acknowledged in the east to be important Church Fathers.)
I could not locate anything that would substantiate the claim that Gregory of Nazianzus (also known as Gregory the Theologian in the east) ever supported anything like what you suggest.
1 Canons against Origen of the Fifth Ecumenical Council
2 p.349n
3 First Principles, III.VI.6
