Theorem: Largest Pandigital Square

The largest pandigital square (in the sense where pandigital includes the zero) is $9 \, 814 \, 072 \, 356$:
:$9 \, 814 \, 072 \, 356 = 99 \, 066^2$

Proof:
We check all the squares of numbers from $99 \, 067$ up to $\floor {\sqrt {9 \, 876 \, 543 \, 210} } = 99 \, 380$, with the following constraints:
Since all these squares has $9$ as its leftmost digit, the number cannot end with $3$ or $7$.
The number cannot end with $0$ since its square will end in $00$.
A pandigital number is divisible by $9$, so our number must be divisible by $3$.
These constraints leaves us with the following $74$ candidates:
{{begin-eqn}}
{{eqn | l = 99 \, 069^2
      | r = 9 \, 814 \, 666 \, 761
}}
{{eqn | l = 99 \, 072^2
      | r = 9 \, 815 \, 261 \, 184
}}
{{eqn | l = 99 \, 075^2
      | r = 9 \, 815 \, 855 \, 625
}}
{{eqn | l = 99 \, 078^2
      | r = 9 \, 816 \, 450 \, 084
}}
{{eqn | l = 99 \, 081^2
      | r = 9 \, 817 \, 044 \, 561
}}
{{eqn | l = 99 \, 084^2
      | r = 9 \, 817 \, 639 \, 056
}}
{{eqn | l = 99 \, 096^2
      | r = 9 \, 820 \, 017 \, 216
}}
{{eqn | l = 99 \, 099^2
      | r = 9 \, 820 \, 611 \, 801
}}
{{eqn | l = 99 \, 102^2
      | r = 9 \, 821 \, 206 \, 404
}}
{{eqn | l = 99 \, 105^2
      | r = 9 \, 821 \, 801 \, 025
}}
{{eqn | l = 99 \, 108^2
      | r = 9 \, 822 \, 395 \, 664
}}
{{eqn | l = 99 \, 111^2
      | r = 9 \, 822 \, 990 \, 321
}}
{{eqn | l = 99 \, 114^2
      | r = 9 \, 823 \, 584 \, 996
}}
{{eqn | l = 99 \, 126^2
      | r = 9 \, 825 \, 963 \, 876
}}
{{eqn | l = 99 \, 129^2
      | r = 9 \, 826 \, 558 \, 641
}}
{{eqn | l = 99 \, 132^2
      | r = 9 \, 827 \, 153 \, 424
}}
{{eqn | l = 99 \, 135^2
      | r = 9 \, 827 \, 748 \, 225
}}
{{eqn | l = 99 \, 138^2
      | r = 9 \, 828 \, 343 \, 044
}}
{{eqn | l = 99 \, 141^2
      | r = 9 \, 828 \, 937 \, 881
}}
{{eqn | l = 99 \, 144^2
      | r = 9 \, 829 \, 532 \, 736
}}
{{eqn | l = 99 \, 156^2
      | r = 9 \, 831 \, 912 \, 336
}}
{{eqn | l = 99 \, 159^2
      | r = 9 \, 832 \, 507 \, 281
}}
{{eqn | l = 99 \, 162^2
      | r = 9 \, 833 \, 102 \, 244
}}
{{eqn | l = 99 \, 165^2
      | r = 9 \, 833 \, 697 \, 225
}}
{{eqn | l = 99 \, 168^2
      | r = 9 \, 834 \, 292 \, 224
}}
{{eqn | l = 99 \, 171^2
      | r = 9 \, 834 \, 887 \, 241
}}
{{eqn | l = 99 \, 174^2
      | r = 9 \, 835 \, 482 \, 276
}}
{{eqn | l = 99 \, 186^2
      | r = 9 \, 837 \, 862 \, 596
}}
{{eqn | l = 99 \, 189^2
      | r = 9 \, 838 \, 457 \, 721
}}
{{eqn | l = 99 \, 192^2
      | r = 9 \, 839 \, 052 \, 864
}}
{{eqn | l = 99 \, 195^2
      | r = 9 \, 839 \, 648 \, 025
}}
{{eqn | l = 99 \, 198^2
      | r = 9 \, 840 \, 243 \, 204
}}
{{eqn | l = 99 \, 201^2
      | r = 9 \, 840 \, 838 \, 401
}}
{{eqn | l = 99 \, 204^2
      | r = 9 \, 841 \, 433 \, 616
}}
{{eqn | l = 99 \, 216^2
      | r = 9 \, 843 \, 814 \, 656
}}
{{eqn | l = 99 \, 219^2
      | r = 9 \, 844 \, 409 \, 961
}}
{{eqn | l = 99 \, 222^2
      | r = 9 \, 845 \, 005 \, 284
}}
{{eqn | l = 99 \, 225^2
      | r = 9 \, 845 \, 600 \, 625
}}
{{eqn | l = 99 \, 228^2
      | r = 9 \, 846 \, 195 \, 984
}}
{{eqn | l = 99 \, 231^2
      | r = 9 \, 846 \, 791 \, 361
}}
{{eqn | l = 99 \, 234^2
      | r = 9 \, 847 \, 386 \, 756
}}
{{eqn | l = 99 \, 246^2
      | r = 9 \, 849 \, 768 \, 516
}}
{{eqn | l = 99 \, 249^2
      | r = 9 \, 850 \, 364 \, 001
}}
{{eqn | l = 99 \, 252^2
      | r = 9 \, 850 \, 959 \, 504
}}
{{eqn | l = 99 \, 255^2
      | r = 9 \, 851 \, 555 \, 025
}}
{{eqn | l = 99 \, 258^2
      | r = 9 \, 852 \, 150 \, 564
}}
{{eqn | l = 99 \, 261^2
      | r = 9 \, 852 \, 746 \, 121
}}
{{eqn | l = 99 \, 264^2
      | r = 9 \, 853 \, 341 \, 696
}}
{{eqn | l = 99 \, 276^2
      | r = 9 \, 855 \, 724 \, 176
}}
{{eqn | l = 99 \, 279^2
      | r = 9 \, 856 \, 319 \, 841
}}
{{eqn | l = 99 \, 282^2
      | r = 9 \, 856 \, 915 \, 524
}}
{{eqn | l = 99 \, 285^2
      | r = 9 \, 857 \, 511 \, 225
}}
{{eqn | l = 99 \, 288^2
      | r = 9 \, 858 \, 106 \, 944
}}
{{eqn | l = 99 \, 291^2
      | r = 9 \, 858 \, 702 \, 681
}}
{{eqn | l = 99 \, 294^2
      | r = 9 \, 859 \, 298 \, 436
}}
{{eqn | l = 99 \, 306^2
      | r = 9 \, 861 \, 681 \, 636
}}
{{eqn | l = 99 \, 309^2
      | r = 9 \, 862 \, 277 \, 481
}}
{{eqn | l = 99 \, 312^2
      | r = 9 \, 862 \, 873 \, 344
}}
{{eqn | l = 99 \, 315^2
      | r = 9 \, 863 \, 469 \, 225
}}
{{eqn | l = 99 \, 318^2
      | r = 9 \, 864 \, 065 \, 124
}}
{{eqn | l = 99 \, 321^2
      | r = 9 \, 864 \, 661 \, 041
}}
{{eqn | l = 99 \, 324^2
      | r = 9 \, 865 \, 256 \, 976
}}
{{eqn | l = 99 \, 336^2
      | r = 9 \, 867 \, 640 \, 896
}}
{{eqn | l = 99 \, 339^2
      | r = 9 \, 868 \, 236 \, 921
}}
{{eqn | l = 99 \, 342^2
      | r = 9 \, 868 \, 832 \, 964
}}
{{eqn | l = 99 \, 345^2
      | r = 9 \, 869 \, 429 \, 025
}}
{{eqn | l = 99 \, 348^2
      | r = 9 \, 870 \, 025 \, 104
}}
{{eqn | l = 99 \, 351^2
      | r = 9 \, 870 \, 621 \, 201
}}
{{eqn | l = 99 \, 354^2
      | r = 9 \, 871 \, 217 \, 316
}}
{{eqn | l = 99 \, 366^2
      | r = 9 \, 873 \, 601 \, 956
}}
{{eqn | l = 99 \, 369^2
      | r = 9 \, 874 \, 198 \, 161
}}
{{eqn | l = 99 \, 372^2
      | r = 9 \, 874 \, 794 \, 384
}}
{{eqn | l = 99 \, 375^2
      | r = 9 \, 875 \, 390 \, 625
}}
{{eqn | l = 99 \, 378^2
      | r = 9 \, 875 \, 986 \, 884
}}
{{end-eqn}}
By inspection, none of these numbers are pandigital.
{{qed}}