Brute-force proof that single-digit numbers are the only square repdigits:
The final two digits of N^2 are determined by the final two digits of N. Checking all 100 possibilities and eliminating the ones that end in two zeros, we find the only remaining options are 12, 38, 62, and 88, all of which end in 44.
So any repdigit square must consist of all 4s. But then it would be 4 times a repdigit consisting entirely of 1s, which (since 4 is a square) would have to be a square itself.
We've already established there are no squares that end in 11, so there cannot be any repdigit squares that con…