No, not all bodies emit Hawking radiation. However, it may be that any body, or spacetime where there is a horizon will produce Hawking radiation.
The explanations for Hawking radiation are various, and equivalent (taking some liberties to simplify things using virtual particle concepts). One way, you have a quantum field outside a horizon in a Schwarzschild (spherically symmetric) black hole BH, and you write the equation for the field in the gravitational metric. After much manipulation and more, the solutions are sums of terms at different frequencies, and positive or negative frequencies which in the coordinate system used represent ongoing and outgoing particles. These exist even in the ground state of the field, i.e., in vacuum. The Ingoing particles go into the BH horizon (have negative mass and reduce the mass of the BH), and the outgoing ones escape. And they calculate that it's thermal blackbody radiation at a temperature determined by constants and the BH horizon size. And the relation to the mass and horizon area, and the entropy which gets to be proportional to the horizon area. Yes, they have to be pretty careful with redshifts etc, and in which frames you see what. At infinity you see the outgoing particles, and that's the Hawking radiation.
The same is true in rotating BHs.
There are no Black Holes and no horizons in a simple planet or Sun. So the same does not work. It seems that it requires a horizon.
The reason there is some thinking that the same happens anytime there are horizons is that they've found other solutions with horizons, and what happens to fields on one side. For instance, it was determined by Rindler, in his solution, that if you have an accelerating particle in vacuum, as it keeps accelerating to closer to c, a back horizon forms. Unruh then found the Unruh effect, and it turns out it emits blackbody radiation, with an equation similar to Hawking's. I have the link to a wiki article on it below, and it has more references. It has another explanation for the effect, but in essence a freely falling observer just sees vacuum and nothing on it, while an accelerated observer sees radiation. The particle number in quantum field theory is not a general relativistic invariant.
If you think about it the Rindler horizon is also the same as the BH horizon (turns out except for extremal BHs) locally, i.e., close to the horizon, and so the two effects are related. There's been other solutions and attempts to generalize the result to any horizon.
I do not know if it generalizes to any horizon but I've read 'popular' articles that say so. Still, I've not investigated the scientific literature fully.
The effect has not been observed, but the Hawking result is widely accepted, Unruh also more or less.
But nothing like that process seems reasonable for just any body. It'll be interesting if LHC discovers any microscopic BHs, because it would then also detect (and it might be the only way to see the BH) the Hawking radiation which for very small BHs, upon extinction, would be a small mini explosion as the BH lifetime is inversely proportional to its mass and horizon size.