According to the Christian Boyer's website, there are three parametric solutions for 4x4 magic squares of squares below.
One of them is Euler's solution.

See Multimagie.com
               
                    sum=85(k^2 + 29)                                          

               
                    sum=65(k^2 + 106)                                          


We searced the solutions for 4x4 magic squares of squares and 11 solutions were found below.


Method

               
                    sum=(a1^2+a2^2+a3^2+a4^2)k^2+b1^2+b2^2+b3^2+b4^2     
                                                         
We consider above 4x4 magic squares of squares.
In order to the 3 rows ,3 columns, and 2 diagonals have the same magic sum, 9 following restrictions are derived.
We obtain the solutions by solving simultaneous equations below.

1. b1^2+b2^2=c3^2+c4^2
2. b3^2+b4^2=c1^2+c2^2
3. b2^2+b3^2=c2^2+c3^2
4. b1^2+b4^2=c1^2+c4^2
5. a1b1+a2b2+a4b4 = a3b3
6. a1c4+a2c3 = a3b3+a4b4
7. a1c4+a4c1 = a2c3+a3c2
8. a1b1+a3c2 = a2c3+a4b4
9. a2b2+a3c2+a4c1 = a1b1



               
                    sum=50(k^2+337)                                          


               
                    sum=130(k^2+281)                                          

     
               
                    sum=125(k^2+373)                                          

               
                    sum=130(2k^2+53)                                          

               
                    sum=145(k^2+74)                                          
             
               
                    sum=340(k^2+29)                                          
             
               
                    sum=290(2k^2+37)                                          
             
               
                    sum=130(k^2+281)                                          
             
               
                    sum=221(k^2+85)                                          
             
               
                    sum=481(k^2+50)                                          
             
               
                    sum=325(k^2+149)