I've heard repeatedly the claim that there is a specific prediction by the Vilna Gaon regarding the dates of Yom HaAtzma'ut (5 Iyar) and Yom Yerushalayim (28 Iyar). What is the source for this claim? Are there other dates in the year that are also predicted there as significant?
As far as I'm aware, there are two sources for the quote.
The first source for this is the book קול התור, allegedly written by Rav Hillel Rivlin M'shklov (one of the Goan's students), some 200 years ago.
The thing is that the published book is an "abridged version", as the initial writings were deemed too complicated. There are actually two published versions (from 1947 and 1968). Interestingly, the passage referring the 5th of Iyar only appear in the 1968 version.
The quote refers to the date the Gaon's students first opened their beit-medrash in Jerusalem. The quote is as follows:Bottom of the page
ויחד עם זה הבנת פעולה חשובה לבנין ירושלם נראה לנו תלמידי רבנו הגר"א בגוונין נהירין כי באותה שעה נפתח החלון הראשון של מחיצת הברזל לחבור הזכות של ברית אבות (החבור שהוא יסוד דתפארת דרך מלכות) שהיה נפסק מחורבן בית המקדש. ואותו היום היה יום ה"עשרים בעמר" שהוא יסוד דתפארת כידוע ליודעי חן
"it's obvious to us, the students of the Gr"a that at this point the first window of the Iron Curtain opened to connect to the "Zchut Avot" (Merits of our Forefathers) that ceased since the destruction of the Temple. And that day was the 20th of the Omer."
The second (and more definitive) source I know of is from מדרש שלמה, written by Rabbi Shlomo Rivlin who published a series of lectures "based on" stuff his grandfather (of the same name) gave dvar torah's about (Grandfather lived ~150 years ago, about 100 years after the Gaon passed away). Both are/were student of the Gaon's school. The date of publication I have for this is 1953 (so after the 1948 war, but before the liberation of Jerusalem).
The quote is as follows ("מדרש שלמה" (דרשה ט"ז ע' 53":
וזהו הטעם שבימי הספירה צריכים להיזהר יותר ממגע עם חברותא רעה ומדברי סכנה, זולת שני ימים מסוימים בימי הספירה שאין הקליפה יכולה לשלוט בהם, והם יום העשרים בעומר, ויום הארבעים ושניים בעומר
And this is the reason that we must be extra careful to refrain contact with evil connections and [things of] danger, except for two specific days in the Sephira [of the Omer] that the "klipa" cannot control [i.e. these are "good days], and they are the twentieth day of the Omer and the Forty-second day of the Omer.
Edit: Thanks to Shamiach for reminding me, the 42nd day of the Omer comes out on the 27th of Iyar, while the official date for the Liberation of Jerusalem is the 28th, so there is an "off by one" issue.
Note that the attribution to the Vilna Gaon is somewhat controversial, and there are some (notably Rav Sternbuch of the Eidah Charaydit) who have greatly disputed this interpretation.