Math, Inc careers page puzzle

Suppose that today is June 1, 2025. We call a date "square" if all of its components (day, month, and year) are perfect squares. I was born in the last millennium, and my next birthday (relative to that date) will be the last square date in my life. If you sum the square roots of the components of that upcoming square birthday (day, month, year), you obtain my age on June 1, 2025. My mother would have been born on a square date if the month were a square number; in reality it is not a square date, but both the month and day are perfect cubes. When was I born, and when was my mother born?

You were born on 25 September 1971.

Your mother was born on 1 August 1936.