--- This is a good question by '巡天剑客', I'm too late to back here. But, my SAS Programm used time 1:36:17 to got 4 solutions. They are as following:
Digit=6 Root=836 Symmetric Square=698896
Digit=12 Root=798644 Symmetric Square=637832238736
Digit=16 Root=64030648 Symmetric Square=4099923883299904
Digit=22 Root=83163115486 Symmetric Square=6916103777337773016196