Sum of all three digit numbers (no digit  being zero) having the property that all digits are perfect  squares, is

The non-zero perfect square digits are 1, 4 and 9 1 can occur at units place in 3 × 3 = 9 ways.

$\therefore$  Sum due to 1 at units place is 1 × 9. Similarly, sum due to 1 at tens place is 1 × 10 × 9 and sum due to 1 at hundreds place is 1 × 100 × 9. We can deal with the digits 4 and 9 in a similar way

Thus, sum of the desired number is

(1 + 4 + 9) (1 + 10 + 100) (9) = 13986