First, let's review Nihon Shoki. It tells us that the very first emperor, Jinmu, took the position on January 1 of a year of Shin Yu. The year name is one of 60 that come and go every 60 years. So, there is a problem in determining which Shin Yu year it is talking.
For example, the year AD1 happens to be a year of Shin Yu. Any year a multiple of 60 years before or after falls Shin Yu. For some reason, writer(s) of Nihon Shoki took the year BC660, as the start year for the emperor Jinmu.
Nowadays, it is widely believed that the year-dating is 660 years off. Or, it is dated too, too old, far back. According to the theory, mathematically, the Jinmu era should have started in AD1 (or +/- 60*n years?). But the mathematical answer has not been confirmed either by any other documents or historical sites or remains.
[Following paragraph updated on 2010/02/02]:
Talking about the January 1 date, this is in the lunar calendar system. The date was interpreted into the solar calendar system, to be January 29, by the Meiji government in 1872. Indeed, January 1 (lunar) coincided to January 29 (solar) for the next year 1873. The government later changed the date to February 11. This was because, the January 1 date in BC660 was February 11 on the Julis calendar of that year.
[The update as above was triggered by a mail from a reader; thank you.]
Since Japanese militarism from Meiji era to 1945, the end of the WW II, was based on the political abuse of emperors, Japanese common people started to view any imperial connotations as bad after the war.
There are still people who are fond of old days and want to see the country's birthday widely celebrated. Others are either indifferent or against the possible revival of the militarism and therefore against celebrating the holiday (yet, taking the holiday,,,.) People with stronger feeling against the militarism hates the national flag and anthem, (except in Sumo events or such international events as Olympic or World something.)
As above, our country's real birthday has not been scientifically determined. February 11 has no foundation, unless one is fond of old days --- only 50 to 130 years old... If one is really fond of old days, he/she should go back, say, 2000 years.
My personal feeling to this problem is that it is nice if we can determine the most likely birthday for our country, but it has to be scientific. Japanese governments have not permitted scientific research of may-be-tomb of emperors or imperial family. In some cases, The Court Agency still keeps a guard at a tomb which has been determined not of the imperial family's. To me, it is impolite to the imperial family if the Agency keeps a tomb of somebody unknown. It's scaring if the Agency advises our Emperor to bow at the tomb of non-royal family.
Well, this is why we take 2/11 off. Thank you for reading.