All the squares of natural numbers possible by taking the product of the
digits of a two-digit number are these: 

0,1,4,9,16,25,36,49,64,81 

since the largest two digit number is 99, which has product of digits 81.  

The numbers 10,20,30,..,90  account for 9 two-digit number which have 0 as
their product of digits.  That's 9.

11 is the only one that has product of digits 1. That makes 10.

14, 22, and 41 are the only ones with product of digits 4.  That makes 13.

19, 33, and 91 are the only ones with product of digits 9.  That makes 16.

28, 44, and 82 are the only ones with product of digits 16.  That makes 19.

55 is the only one with product of digits 25.  That makes 20.

49, 66, and 94 are the only ones with product of digits 36.  That makes 23.

77 is the only one with product of digits 49.  That makes 24.

88 is the olny one with product of digits 64.  That makes 25.

99 is the only one with product of digits 81.  That makes 26.

Answer: 26.

Edwin