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On the first square root day of this century, 1/1/01 (if you'll forgive the leading zero) I visited the Greenwich Observatory and, as one does, stood astride the 00:00:00 longitude line. I actually didn't put the coincidence together until after the time was 13:01:01.

Nice that this works whether you prefer UK or US date format.

Nine. (But if you took the last three digits of the year instead of the last two, this century would be special with ten fitting dates...)

Five more should come in 21st century. Next is 6/6/36. And this year is a square root year as well. Last was 1936 and next square root year will be 2116.