It has been mathematically proven that the smallest number in a complete 3x3 magic square of squares – if it even exists – would have to be larger than 10^14.

I would definitely recommend starting with the algebra before moving on to brute force. Personally, I've found a parametric formula which turns 4 variables of almost any arbitrary value into a 5/9 square of squares (the 4 corners and the center), and I've brute forced about 150 specific values which create 6/9 squares (the 4 corners, the middle, and one of the sides).

Out of all of the 6/9 squares that I have found myself, my favorite (per my love of horrifyingly large numbers) is the one with

17-digit square, 14-digit square, 18-digit square
18-digit non-square, 17-digit square, 15-digit non-square
14-digit square, 18-digit non-square, 17-digit square

However, I have not found a general parametric formula for generating 6/9 squares from any arbitrary value of variables, nor have I found any 7/9 squares by applying brute force to my 5/9 formula.