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As far as I know you put dates in Genitive case, when you are saying something occurred on that date. So, for example
We met on the twenty-second of June.
Would be
Мы познакомились двадцать второго июля.
But a grammar textbook I have is saying that the following should not be in Genitive and I can't understand why, when it is saying something happened on that date.
My birthday - the twenty first of December.
Мой день рождения - двадцать первое ноября.
Both Old Russian and Church Slavonic used genitive to mark a moment in time.
This has survived in Modern Russian in constructs like первого числа, второго (дня месяца) октября, третьего дня ("three days ago"), четвёртого года ("four years ago"), and has been fossilized in words вчера, сегодня, завтра, днесь, all originating from ancient genitive forms.
So if the date answers the question "when", like in "when did it happen", you should put it into genitive:
- When did you meet? We met on July, 22th // Когда вы познакомились? Мы познакомились двадцать второго июля.
However, if the date answers the question "what", like in "what day was that", you put it into nominative or whatever case is required grammatically to answer that:
What is the date were you born on? That date is November, 21st. // Какая дата твоего рождения? Дата моего рождения -- двадцать первое ноября.
By what date should it be done? This should be done by March 1st // К какой дате это нужно сделать? Это нужно сделать к первому марта.
Those two can easily be confused, as "what date" and "when" are very close in meaning. So both those answers are in fact valid:
When is your birthday? My birthday is (when?) on November, 21st. // Когда у тебя день рождения? У меня день рождения (когда?) двадцать первого ноября.
What is your birth date? It's (what?) November 21st. // Какой день твоего рождения? День моего рождения — (что?) двадцать первое ноября.
The case depends on what the dash stands for (even in English "My birthday - the twenty first of December" the dash implies "is on", otherwise the sentence in ungrammatical). In Russian you could imply "был/будет" ("was on/will be on", when you talk about a particular occurrence), in such a case it would be genitive case:
Мой день рождения [был/будет] двадцать первого ноября.
But in your example the dash stands for something like "приходится на" ("falls on", i.e. every year), and it is an accusative case (винительный падеж, because it answers the question "приходится на кого/что?"):
Мой день рождения [приходится на] двадцать первое ноября.
Update: the dash could also imply "это", i.e.:
Мой день рождения [это] двадцать первое ноября.
thus being a plain nominative case (answers the question "это кто/что?"). But this variant sounds less literate to me.
Update2: suddenly realized that the "accusative" part above is incorrect: while it's possible to say "Мой день рождения приходится на двадцать первое ноября.", it's impossible to substitute "приходится на" with a dash here, so this variant doesn't apply to the original question, and the rest of my answer boils down to what Quassnoi explained in greater detail.
