My friend has taken the challenge of the OP and asserts that 1 and 81 are the only positive integers with that property. (Admitting 0 makes for an additional solution.)

His proof (sketched): assume the number to have n decimal places, and x being its square root (both positive integers), then 10ⁿ ≤ x². Since x is the sum of those decimal digits, x ≤ 9n. Everything being positive, we can square this to get x² ≤ 81n². Together, we get 10ⁿ ≤ 81n². This is not true for n=3 (1000 being greater than 729), and it cannot be true for any higher n, roughly because the left side grows exponentially. ("The details are left to the reader.") Thus n can only be either 1 or 2, easily seen to yield no more than the two solutions mentioned.

All school maths, nothing esoteric.