Hermite's theorem  indeed generalizes to entire functions but your statement for entire functions is incorrect. 
$$2\cos2z-\cos  z=4\cos^2z-1-\cos z$$
has all zeros real, but the zeros of $2\cos 2z$ and $\cos z$ do not interlace.
Substitute the square root if you want zeros to be on a ray.

The correct statement: If $f$ and $g$ are real entire functions, and 
$f-ag$ has
only real roots FOR EACH REAL $a$, then
zeros of $f$ and $g$ are interlacent.

Proof. The first statement is equivalent to saying that $f/g$ has imaginary part
of constant sign in the upper half-plane and the opposite sign in the lower
half-plane. The family of such functions in normal in each half-plane.
So it is enough to prove this for polynomials, and for polynomials it is easy.

For the converse to be true, the you need a priori assumptions on your functions,
like the assumption that you make that they are of genus 0, with zeros on a ray.
For a complete discussion of these questions the reference is Levin, Distribution of zeros of entire functions.