Are there any particular dates when Moshiach's coming is more likely than other times?
Moshiach can come at any time, but certain days are considered especially auspicious times for redemption.
In the words of the Lubavitcher Rebbe, the present era is "high time" for Moshiach's arrival. We anticipate the redemption to begin at any moment.
However, there are certain days that have always been considered as especially fitting days for Moshiach's arrival. For example, the entire month of Nissan, as our sages say: "In the month of Nissan our ancestors were redeemed from Egypt, and in Nissan they will be redeemed in the future."
The 23rd of Sivan similarly is considered an especially auspicious day for redemption. The Scroll of Esther relates that on the 23rd of Sivan, King Ahasuerus ordered his scribes to issue an edict giving the Jews under his domain the right to defend themselves against their enemies. On that same day, “Mordecai left the king's presence with royal raiment... the city of Shushan shouted and rejoiced, and the Jews had light and joy, and gladness and honor.”
The special arousal of divine favor that the Jews experienced on the 23rd of Sivan recurs every year on this date. If we only utilize it, it can become the date of the greatest good tidings of all - the true and complete redemption.
The “King's scribes” are the angels and ministers of the King of the universe - G-d Almighty. On the 23rd of Sivan, the angels grant the Jewish people extraordinary strengths to fulfill the mitzvot and feel “light, joy, gladness and honor” in the innermost sense of the word. “Light is Torah, joy is the holidays, gladness is the mitzvah of circumcision, and honor is the mitzva of tefillin.” If we only utilize the power inherent in this day, the Geulah will come immediately.