The Chol/Avarshina.
First is a source commenting on the pasuk in Iyov regarding a bird known for its longevity, the "Chol," as the pasuk (29:18) says, "וכחול ארבה ימים, And like the Chol I will multiply days." The below is quoted from Yalkut Shimoni on Iyov (29, Remez 917), which I have also seen quoted from Bereishis Raba 19:
רבנן אמרין כיון שאכלה חוה פרי האילן האכילה את הבהמה ואת החיה ואת העופות הכל שמעו לה חוץ מעוף אחד ושמו חול דכתיב וכחול ארבה ימים, דבי רבי ינאי ורבי יודן ברבי סימון, דבי רבי ינאי אמרי אלף שנים הוא חי ולסוף אלף יוצאה אש מקנו ושורפתו ומשתייר בו כביצה וחוזר ומגדל אברים וחיה
The sages say, when Chava ate from the fruit of the tree, she fed it to the domesticated animals, the wild animals, and to the birds. Everyone listened to her except for one bird which is called "Chol," as the pasuk (Iyov 29:18) says, "And like the Chol, I will multiply days." [this was taught by] the house of Rabbi Yanai and Rabbi Yudan in the name of Rabbi Simon. The house of Rabbi Yanai says, "It lives for a thousand years and at the end of a thousand, a fire emerges from its nest and burns it up. All that is left from it is something the size of an egg. It returns and grows limbs and lives."
This source seems the closest to the version of the Phoenix story I'd heard before. It makes sense that the Midrash says that because the Chol did not eat from the fruit of the tree of good and evil, it lives indefinitely. Death is not a natural state. As long as creation was not touched by sin and was connected to Hashem, the source of life, there was no reason why anyone should die. When sin entered the world, creation cut itself off from the source of life. The automatic result is death. Since the Chol never ate from the fruit and did not sin, it is not subject to the curse of death.
Interestingly, sefer Iyov (the source of the Midrash's reference to the Chol) was written by Moshe Rebbeinu (Bava Basra 15a), which is a source predating the time of the Greeks. It should be noted that according to some meforshim on the pasuk in Iyov, however, this refers to a bird with a long life, but not one which burns up or dies and regenerates itself.
The Gemara in Sanhedrin (108b), however, tells a different version related to Noach:
אורשינה אשכחיניה אבא דגני בספנא דתיבותא א"ל לא בעית מזוני א"ל חזיתיך דהות טרידא אמינא לא אצערך א"ל יהא רעוא דלא תמות שנאמר ואומר עם קני אגוע וכחול ארבה ימים
[Shem, the son of Noach said:] My father found the Avarshina [the Gemara's word for the Chol - Rashi] hiding in its quarters in the ark [instead of going out to eat like the other animals]. He [Noach] said to it, "You do not require food?" It said to him, "I saw that you were preoccupied. I thought that I would not trouble you." He said to him, "May it be His will that you should not die," as the pasuk [in Iyov] says, "I will perish with my nest, and like the Chol I will multiply days."
According to this Gemara, it sounds like the bird merits an extraordinary long life because of Noach's blessing. The Maharsha points out that this seems to contradict the Midrash quoted above, which indicates that the bird merited long life from the time of Chava. According to the Artscroll Gemara, the Maharal and other meforshim offer various pshatim to explain why this is not a contradiction. If I remember correctly, the Maharal says that Iyov was merely reaffirming the bird's existing blessing.
The "Eagle."
The Radak discusses a somewhat similar bird based on a pasuk in Tehillim (103:5). The pasuk says "תתחדש כנשר נעוריכי, renew your youth like an eagle." I can't find the Radak online in Hebrew so I'll just translate what he says about the nature of this "eagle:"
They say that the eagle renews itself every ten years. It ascends very high to the face of the firmament of the heavens and when it approaches the heat of the primordial fire, it causes itself to fall into the sea because of the great heat. And it dies and renews itself and regrows its limbs and returns as in the days of its youth. It does this every 10 years until one hundred. And when it reaches one hundred, it ascends in its usual way and causes itself to drop into the sea and dies there.
This version is markedly different from the Chol/Avarshina bird discussed in the Yalkut Shimoni/Midrash Raba and the Gemara in Sanhedrin so I'm not 100% sure it's talking about the same creature. In addition, this version of the story sounds like an amalgam of the Phoenix and
Icarus stories.
Bottom line.
Whether the Chol, Avarshina, unique Eagle, or the Phoenix actually exist or existed historically or zoologically is really irrelevant. Similarly, whether the Greeks knew about this bird from Moshe in sefer Iyov or had a separate tradition from the same source Iyov/Moshe got it from, is also irrelevant. The story clearly resonates with people because it carries a truth deeper than its historicity.
The real point of chazal's and Dovid Hamelech's references to this creature is that no matter what a Jew has been through or done, he can always begin again. This is true on a national level and on an individual level. I wrote
here and in
this follow-up post how (similar to the Phoenix), the fundamental truth is that one must first destroy the old in order to build something new and greater in its place. That is personified by the Phoenix, but it is true for us as well.
IY"H, may we all be zoche to recreate ourselves from scratch so that we will never be weighed down by the baggage of the past!
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We are putting together a men's Krav Maga class two mornings a week in the social hall at Aish Kodesh so we can get healthy and learn some self-defense as well. It would be taught by Eilon Even-Esh, a frum fellow who served as a captain in the Marines and as a staff seargent in the IDF, where he served as a special forces soldier in the Dubdevan and Oketz units. During his career, he has studied Krav Maga, Muay Thai, Ju (Jew?)-Jitzu, kickboxing, and in the Marine Corps Martial Arts Program. Eilon also teaches Krav Maga to individuals and other groups so if you would like to set something up for yourself somewhere else or at a different time, feel free to get in touch with him using his contact information on his website.
The Aish Kodesh morning class would be Tue. and Thu .mornings from about 7:45-8:45, right after the 6:55/7:05 minyan. It will therefore be over in time to catch the 9:06 train. It would take place in the social hall and the mechitza or dividing wall would be set up for privacy. 
Because of Eilon's and his wife's special connection to Aish Kodesh, we have a special price which I can discuss with anyone who gets in touch with me (don't want to put it on the blog). In order to get that rate, we'll need a minimum of 8 guys. 
