Let John's age be $10 a+b$ where $0 \leq a, b \leq 9$. Then Mary's age is $10 b+a$, and hence $a>b$. Now

$$(10 a+b)^{2}-(10 b+a)^{2}=9 \cdot 11(a+b)(a-b) .$$

Since this is the square of an integer, $a+b$ or $a-b$ must be divisible by 11 . The only possibility is clearly $a+b=11$. Hence $a-b$ must be a square. A case study yields the only possibility $a=6, b=5$. Thus John is 65 and Mary 56 years old.