Consider the point $ (x,y) $. Then, he labels the point with the number $$(\sqrt{(x-0)^2 + (y-0)^2})^2 = x^2 + y^2,$$ so it follows that $ x^2 + y^2 = 25 $. From here, some casework needs to be done to find the number of pairs $ (x,y) $ that satisfy $ x^2 + y^2 = 25 $. We note that $ x^2 = 25 - y^2 \le 25 \Longrightarrow |x| \le 5 $, and so $ |x| $ can only be equal to $ 0,1,2,3,4,5 $. Of these, only $ 0,3,4,5 $ produce integer solutions for $ |y| $. If $ |x| = 3 $, then $ |y| = 4 $, and any of the four combinations $ (3,4)(-3,4)(3,-4)(-3,-4) $ work. Similarly, if $ |x| = 4, |y| = 3 $, there are four feasible distinct combinations. If $ |x| = 0 $, then $ |y| = 5 $, but then there is only one possible value for $ x $, and so there are only two combinations that work: $ (0,5) $ and $ (0,-5) $. Similarly, if $ |x| = 5, |y| = 0 $, there are two feasible distinct combinations. In total, there are $ \boxed{12} $ possible pairs of integer coordinates that are labeled with $ 25 $.