Getting back to the problem, I have found something interesting. 7^n can only end in the numbers 9,3,1,7 and 24^n can only end in numbers 4,6. And this happens in predictable ways ! 24^n ends in 4 when n is odd and 6 when n is even. Also for 7^2 you can figure out the ending number based on n mod 4. Also any number squared can only end in 0,1,4,5,6,9. So if you look at different possibilites for what the ending number of 7^n+24^n is under different n mod 4s, you can see if that number is even possible to be a perfect square based on its last digit. It eliminates a lot of possibilities, but I have't found a way to find real solutions.
Also btw to the other commentors, I think the 25 is the number that is squared (and thats why it is included in the answer). [MATH] 7^2+24^2=25^2[/MATH]