x² - y² = 133

(x - y)(x + y) = 133

So (x - y) and (x + y) are factors of 133
and (x - y) is the smaller factor and (x + y) 
is the larger factor. 

The only two ways to factor 133 are

1·133 where 1 is the smaller and 133 is the larger.  

and

7·19  where 7 is the smaller and 19 is the larger.

In the first case we have

x - y = 1
x + y = 133

Solve that system and get solution x=67, y=66

The sum of their squares is 67²+66² = 4489 + 4356 = 8845 

--------------

In the second case we have

x - y = 7
x + y = 19

Solve that system and get solution  x=13, y=6

The sum of their squares is 13²+6² = 169 + 36 =  205

Edwin