EDIT: I missed that the question asked for the digits 1 through 9 used exactly once. My logic works for a date that has numbers 1 through 9 used no more than once.
Ensqare was born on
April 25, 81 at 9:36
Reasoning:
First let us get which numbers qualify for the month, hours, days, and minutes. That is a count of 7 (1,4,9,16,25,36,49) numbers that need to fill 4 spots.
I then went the long way around and listed all squares from 1 to 2021. First I removed any number that had a repeated digit or a number that contained a 0, this left us with 27 numbers to choose from for the year.
Looking at the 7 numbers for the min/hr/day/mnth spots, we can see that if the year contains a 6 and a 9, that would eliminate 4 of our 7 numbers (9, 16, 36, 49) and we wouldn't have enough left to populate our min/hr/day/mnth spots. Similarly, a year containing a 4 and a 6 would eliminate 4 numbers (4, 16, 36, 49). Similarly (again), if the year contains a 4 and a 9 that would eliminate 3 numbers (4, 9, 49) and then 16 would clash with 1 and 36. This leaves us with 17 years to choose from.
From here we can start checking numbers. If there are less than 4 valid numbers, it is a bust and we can throw it out. I don't know how to make a table, so this is the best I can do.
Year | valid numbers (1, 4, 9, 16, 25, 36)
841 | 9, 25, 36 (bust)
784 | 1, 9, 16, 25, 36
729 | 1, 4, 16, 36 (bust, 36 and 16 clash)
625 | 1, 4, 25, 49 (bust, 4 and 49)
576 | 1, 4, 9, 49 (bust, 4 and 49)
529 | 1, 4, 16, 36 (bust, 1 and 16)
361 | 4, 9, 25, 49 (bust 4, and 49)
324 | 1, 4, 9, 16, 49 (bust, 4 and 49/1 and 16)
289 | 1, 4, 16, 36 (bust, 1 and 16)
256 | 1, 4, 9, 49 (bust, 4 and 49)
81 | 4, 9, 25, 36
36 | 4, 9, 25, 49 (bust, 4 and 49)
25 | 1, 4, 9, 16, 36, 49
16 | 4, 9, 25, 49 (bust, 4 and 49)
9 | 1, 4, 16, 25, 36
4 | 1, 9, 16, 25, 36
1 | 4, 9, 25, 36, 49
This leaves us with the years 784, 81, 25, 9, 4, 1 as possible solutions
From there I calculated valid dates starting with the largest year, and then checked their day of the week at timeanddate.
For year 784, the number 36 has to be the minutes (it is too large for month, day, or hour) and 25 has to be the day (too large for month or hour).
year | mth | day | hr | min
784 | 9 | 25 | 1 | 36 (Saturday)
784 | 1 | 25 | 9 | 36 (Sunday)
For year 81, the number 36 has to be the minutes and 25 has to be the day from similar logic as above.
year | mth | day | hr | min
81 | 9 | 25 | 4 | 36 (Thurs)
81 | 4 | 25 | 9 | 36 (Wednesday)
April 25, 81 at 9:36 was the first occurrence of a Wednesday so that is where I stopped
Note: If anyone want's to format my tables to be prettier, I would greatly appreciate it