There seems to be a difference of opinion, unexpressed, as to what we mean when we say 'free particle'. It has to mean, at the very least, that it does not interact with any potential energy term. We can, then, consider it as part of a closed system in which there is no potential energy, so its Hamiltonian has no potential energy term. In fact, we usually mean to say that its Hamiltonian has the usual form from quantising a classical free Hamiltonian, and it is clear that that is what Mr. Eichenlaub, and some other contributors, meant. (The web interface here for some ungodly reason turns my backslash into yen...so I merely refer to the Hamiltonian written above.)
Now no such particles exist in Nature for the trivial reason that there are, indeed potential energy terms all over the place, no particle is really isolated from the rest of the universe. So the question is about a hypothetical situation, and is a reasonable question.
A free particle cannot be in an eigenstate of the Hamiltonian because the Hilbert space of states does not possess any such eigenstates: this is simply another way of saying what Mr. Eichenlaub wrote, and it cannot be really criticised. The next step is that if it had a definite energy it would be an eigenstate, so that is why it cannot have a definite energy.
It is rather unreal to first criticise this line of reasoning by saying one must look at reality only, since really there are no free particles at all. But it is then positively inconsistent to go on and talk about experimentally observed free particles.
Obviously there can be particles which are approximately free, but then, they could have a wave function that was approximately a plane wave: i.e., a narrow-band superposition in which their position would have a very large variance (but not infinite), and so the probabilities of finding it in any very small location would be practically zero (but not exactly zero). What would be the limits of this approximation? We would have to be justified in neglecting the potential energy, of which there are chiefly two kinds to worry about : forces exerted by other particles, and gravity. So if the particle were far away from all other particles, this might be justified. But the larger the variance of its position, the harder it is to arrange this...
I know that the practicalities of the existence of a free particle were not what the questioner was asking about, but anyone who wishes to make reality trump the formulation of a simple, sensible question about the free Hamiltonian is obligated to come to grips with reality and analyse the limits of this kind of approximation. If my tentative analysis in the previous paragraph is correct, this kind of approximation is justified when we are dealing with a more or less monochromatic wave....
The situation of a monochromatic wave is not what you might think. It is stationary, so nothing is moving. When regarded as a wave, it doesn't change its position, it is not moving that way. When regarded as a particle, it is not moving either. That is the whole point of being in a stationary state... and is the difference between phase velocity and particle velocity.