I've just been playing with things for a bit, and just thought I'd dumb things down for simplicity, which could also mean I dumbed things down a little too far.
If we look at the exponents only, and take the addition of them we have a function
f(x,y)=x^2+y+y^2+x
=\left( x+\frac{1}{2} \right)^2+\left( y+\frac{1}{2} \right)^2-\frac{1}{2}
So we clearly have a symmetry about y=x (as we already knew) and the minimum value f can take is -1/2.
At this point (x,y)=(-1/2,-1/2) which is where the minimum of f occurs, we found that
g(x,y)=16^{x^2+y}+16^{x+y^2}=1
And so if we take some other value, say, x=\frac{-1}{2}+k, y=\frac{-1}{2}-k, k\neq0 ***
then with these values we have
x^2+y=\left(k-\frac{1}{2}\right)^2-\frac{1}{2}-k
=\left(k-1\right)^2-\frac{5}{4}
y^2+x=\left(k+\frac{1}{2}\right)^2-\frac{1}{2}+k
=\left(k+1\right)^2-\frac{5}{4}
Therefore, g becomes
g(k)=16^{(k-1)^2-5/4}+16^{(k+1)^2-5/4}
=\frac{1}{32}\left(16^{(k-1)^2}+16^{(k+1)^2}\right)
And for all k
16^{(k-1)^2}+16^{(k+1)^2}>16+16=32
So we only have the minimum at (-1/2,-1/2) which means this is the only combination (x,y) that satisfies g(x,y)=1
*** I'm uneasy about making this assumption that if we take some small linear increase of k in one variable, we should decrease the other variable with the same magnitude k.
(missed the ^) 
