An odd number times an odd number is an odd number.

An even number times an even number is an even number.

So, for a number times itself to be odd, the number
multiplied by itself to give that number must be odd.

The smallest odd number which times itself gives a three
digit number is 11x11 = 121.  The largest is 31x31 = 961.

Hey!  That's one of them!  I just noticed that the sum of the
digits of 961 is indeed 9+6+1 = 16.  

Are there any others?

Let's find out:

11x11 = 121, sum of the digits is 4, not 16.
13x13 = 169, aha! 1+6+9=16.  That's a solution!
15x15 = 225, sum of the digits is 9, not 16.
17x17 = 289, sum of the digits is 19, not 16.
19x19 = 361, sum of the digits is 10, not 16.
21x21 = 441, sum of the digits is 9, not 16.
23x23 = 529, aha! 5+2+9=16.  That's a solution!
25x25 = 625, sum of the digits is 13, not 16.
27x27 = 729, sum of the digits is 18, not 16.
29x29 = 841, sum of the digits is 13, not 16.
31x31 = 961, that's the first one I found.

So there are three such numbers 169, 529 and 961.

Edwin