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Proposed choice of date:

Short answer: November 5th

Long Answer: I'm going to stick to nonleap years for a moment. So exactly one day of the week repeats 53 times. In 2009 this is a Thursday, so I'm going to say Thursday from there on out. If it is a different nonleap year, just replace Thursday by whatever day of the week happens on Jan 1st that year. You will get the same answer for the date.

Since every month has 4 or 5 of Thursdays, there exactly 7 months which have 4 Thursdays. So at least one quarter of the year has one or less 4-Thursday month. It turns out this is uniquely the 4th quarter, and the month is November. Further, it can't be November 1st or 2nd, because their days of the week happen 5 times. So there must be exactly one Thursday with a single digit date. This turns out to be November 5th.

November 5th is also the first time Thursday occurs on the same day of the month for the third time (February 5th and March 5th were also Thursdays.) And it's the farther Thursday in November from Thanksgiving.
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Pigeon-hole Day

Festivities include talking about proofs or arguments that use the pigeon-hole principle (I like that there are lots of these which are accessible). I'm sure there are also games you could come up with (e.g. n+1 people take turns throwing a coin at n jars. If you miss, you go to the back of the line. First one to put the second coin in a jar is eliminated and one jar is removed).

Food: not pigeons please. Maybe a cake with more than n interesting features being cut into n pieces.

One candidate for the day would be every blue moon (whenever there are two full moons in a month), but this only happens once every three years or so. Another idea is to try using the fact that every day of the week occurs 52 times in a year, except one, which occurs 53 times; I'm not sure of the best way to narrow it down from there.