There are 87 square numbers that use every digit (base 10, no leading zeroes)) exactly once.
(In reply to re: Pencil and Paper solution (spoiler)
Interesting question, broll.
A necessary condition for a number to be pandigital is that it must = 0 mod 9. Well, only 1/9 of all integers equal 0 mod 9, but fully 1/3 of all squares equal 0 mod 9. So I would not be surprised if there were 67 pandigitals in the range.
Maybe the difference between 67 expected and 87 actual is not significant.