A number is
99066
When squared it becomes
9814072356
Explanation
I cutdown the possible answers by doing the following:
Determined the min strobogrammatic number to form a 10 digit number which was 3???? This gets rid of 1,2 as a leading digit (down from 210 to 140 possible answers)
The pandigit number is divisible by 9 (sum of digits is divisible by 9) and if N^2 is divisible by 9 then N is divisible by 3 (and the sum of digits of N is divisible by 3
Each number (x) in the center adds x mod 3 to the total of the digits so we have
0 mod 3 = 0
1 mod 3 = 1
2,5,8 mod 3 = 2
Each number (2) in either of the outside numbers adds 2x mod 3 to the total (or in the case of 6,9 it adds 15 mod 0)
0*2 mod 3 = 0
6+9 mod 3 = 0
9+6 mod 3 = 0
2*2 mod 3 = 1
5*2 mod 3 = 1
8*2 mod 3 = 1
1*2 mod 3 = 2
So we need to choose 2 of the outside digits (only can choose one 0) and 1 of the inner digits
At this point I just tried a few numbers (and happily stated at high numbers) although to carry it through for a final possibility of numbers
Regex shows 1,2,3rd numbers, 4th and 5th are set based on the first 2 numbers
+----------------+---------------+
| Regex | Possibilities |
+----------------+---------------+
| [58][069][258] | 18 |
| [58][258]1 | 6 |
| [58]10 | 2 |
| [69][069]0 | 6 |
| [69][258][258] | 18 |
| [69]11 | 2 |
+----------------+---------------+
Gives a total of 52 possible inputs.
Which is still a fair number to try by hand and I'll admit I didn't realize no computer also meant no calculator until this morning, I thought it was mainly to prevent coded brute force solutions.