Can Someone Who Is Familiar With The Ancient Greek Calendar Help Us Translate Epicurus' Birthday Into the Correct Day To Observe in 2020? Here is the information from Gassendi's Life of Epicurus - Chap 2. The Time of His Birth:
Epicurus was born (as Laertius relates out of the Chronology of Apollodorus) in the 3rd year of the 109th Olympiad, the 7th day of the month Gamelion; at whose birth,  Pliny saith, the Moon was twenty daies old. Hecatombeon (the first month) this year falling in the Summer of the year 4372. of the Julian Period, (now used by Chronologers) it is manifest, that Gamelion the same year, being the 7th month from Hecatombeon, fell upon the beginning of the year 4373, which was before the ordinary computation from Christ 341 compleat years. Now forasmuch as in January, in which month the beginning of Gamelion is observ’d to have fallen, there happened a new Moon in the Attick Horizon, by the Tables of Celestiall Motions, the fourth day, in the morning, (or the third day, according to the Athenians, who as Censorinus saith, reckon their day from Sun-set to Sun-set) and therefore the twentieth day of the Moon is co-incident with the three and twentieth of January; it will follow, that Epicurus was born on the 23rd of January, if we suppose the same form of the year extended from the time of Cefar, upwards. And this in the old style, according to which the cycle of the Sun, or of the Dominical letters for that year, (it being Biffextile) was BA, whence the 23rd day of January must have been Sunday. But if we suit it with the Gregorian account, which is ten daies earlier, (now in use with us we shall find, that Epicurus was born on the 2nd of February, which was Sunday, (for the Dominicall Letters must have been ED.) in the year before Christ, or the Christian computation, 341. and consequently in the 1974th year, compleat, before the beginning of February this year, which is from Christ 1634. Some things here must not be passed by.
First, that  Laertius observes Sosigenes to have been Archon the same year, wherein Epicurs was born, and that it was the 7th year from the death of Plato. Moreover, it was the 16th of Alexander, for it was, as the same  Laertius affirms, the year immediately following that, in which Aristotle was sent for to come to him, then 15 years old.
Secondly, that  Eusebius can hardly be excused from a mistake, making Epicurus to flourish in the 112th Olympiad; for at that time, Epicurus scarce had pass’d his childhood, and Aristotle began but to flourish in the Lyceum, being returned the foregoing Olympiad out of Macedonia, as appears from  Laertius.
Thirdly, that the error which is crept into  Suidas, and hath deceived his Interpreter, is not to be allowed, who reports Epicurus born in the 79th Olympiad. I need not take notice, how much this is inconsistent, not onely with other relations, but even with that which followeth in Suidas, where he extends his life to Antigonus Gonotas: I shall onely observe, that, for the number of Olympiads, Suidas having doubtlesse set down ςθ, which denote the 109th Olympiad, the end of the ς was easily defaced in the Manuscript, so as there remained onely ο, by which means of οθ, was made the 79th Olympiad.
Fourthly, that it matters not that the Chronicon Alexandrinum, Georgius Sincellus, and others, speak too largely of the time wherein Epicurus flourished, and that we heed not the errous of some person, otherwise very learned, who make Aristippus later then Epicurus, and something of the like kind. Let us onely observe what  St. Hierom cites out of Cicero pro Gallio; a Poet is there mentioned, making Epicurus and Socrates discoursing together, Whose times, saith Cicero, we know were disjoyned, not by years, but ages.
Fifthly, that the birth-day of Epicurus, taken from Laertius and Pliny, seems to argue, that amongst the Athenians of old, the Civill months and the Lunary had different beginnings. This indeed will seem strange, unlesse we should imagine it may be collected, that the month Gamelion began onely from the full Moon that went before it; for, if we account the 14th day of the Moon to be the first of the month, the first of the Moon will fall upon the 7th of the month. Not to mention, that Epicurus seems in his Will to appoint his birth to be celebrated on the first Decad of the dayes of the month Gamelion, because he was born in one of them; and then ordaineth something more particular concerning the 20th of the Moon, for that it was his birth-day, as we shall relate hereafter. Unlesse you think it fit to follow the  anonymous Writer, who affirms, Epicurus was born on the 20th day of Gamelion; but I know not whether his authority should out-weigh Laertius. Certainly, many errours, and those very great, have been observed in him, particularly by Meurfius. I shall not take notice, that the XXXX of Gamelion might perhaps be understood of the 20th of the Moon, happening within the month Gamelion, from Cicero, whose words we shall cite hereafter. But this by the way.
You can see next years here http://www.numachi.com/~ccount/hmepa/numachi.com
The day of moon calendar begins from the afternoon so 7th gamelion is from 2 until 3 January on this year. For 2021 the 7th gamelion is from afternoon of 20 January until next afternoon 21 January.
Γᾰμηλῐών • (Gamēliṓn) (genitive Γᾰμηλῐῶνος)
Τhe seventh month of the Attic calendar, corresponding to the lunar term around January and February during which many weddings were customary, because it was the anniversary of Zeus and Hera's weddings but also because wars usually stopped when the winter began. Thus, the couple were enjoying themselves as newly married but also seeking to acquire children - also this period had several celebrations and there were not many agricultural works.
Thank you Takis and Elli! I am going to go ahead and mark our calendars for January 2, 2020, and January 20, 2021! Thank you!