Output of "x^2 + y^2" from 2's complement to binary (with sign)

No, that is not the case.

when adding two N bit wide numbers the largest possible result is N+1 bit wide.

let's assume both x² and y² are the largest possible 16 bit numbers (when using signed arithmetics):

0x7FFF and 0x7FFF

or binary: 0111.1111.1111.1111

Adding those two numbers is equivalent to multiplying one of the numbers with 2.

Multiplikation with 2 again is equivalent to shifting the number to the left by one, so we end up with:

0.1111.1111.1111.1110

which equals 0x0000FFFE

When using unsigned arithmetics from now on we will not need the zeros at the beginning: so we end up with unsigned 0xFFFE.