Why can't a particle rotate about its axis? When treating an atom as a point partcle, it is usually assumed that it cannot rotate about its axis. Is it that anything considered as a point particle cannot rotate about its own axis? If so, why is that the case?
 A: It is because of quantization of the angular momentum. If an atom rotates, that would mean a higher orbital angular momentum for the electrons. Usually, that is a state several electron-volts higher than the ground state. Not accessible at ordinary temperatures.
A: The aim of this answer is to give a physical intuition without quantum mechanics (The answer by Pieter should be the correct rigorous answer):
A monoatomic gas contains per definition of single atom particles. That means these particles consist of the core of the Atom (with the whole mass, roughly point particle in comparison to the size of an atom) and the electronic orbitals (Contain a negligible amount of mass compared to the core).
I think the intuitive way of thinking is to say, by 'rotating' this negligible small mass (the electrons) you are nowhere near the energy scale of the displacement degrees of freedom (DOF) of the atom.
In comparison in a two atomic gas, only if the two atomic cores are rotating around the rotation axis the DOF is considered. We have the same case as for the one atomic gas if both cores lie on the rotation axis.
Hope this gives you some intuition behind this consideration in stat. phys. .
A: If the particle is charged, then rotation would mean acceleration of charge. Rotation means constantly changing the direction of the velocity, ergo acceleration. An accelerating charge would radiate (emit photons), so this state would never be stable. The idea of spin came about because we observed stable charged particles that interacted with magnetic fields as if they were rotating, but weren't emitting any photons. So this classical interpretation led to a contradiction, and we knew it couldn't be the correct picture.
In general, small things in classical theories can rotate. But that's just not how particles actually behave.
A: I will answer the title

Why can't a particle rotate about its axis? for elementary particles.

If you look at the table, almost all elementary particles are axiomatically assigned spin. Spin in classical mechanics is a rotation about an axis going through the center of mass of the particle. Thus , in a classical picture elementary particles do rotate.
BUT elementary particles are quantum mechanical entities, and their position in space is controlled by a probability distribution, which is predicted by the wavefunction of the solution for the boundary conditions of the specific problem, scattering or decay. So what does spin mean for elementary particles?
Spin had to be assigned in elementary particle interactions, particle by particle in the table, in order fot the law of angular momentum conservation to apply at the quantum mechanical frame.
The assignment in the tables has been found to be consistent with all the up to now data. Thus the answer for elementary particles to the title is also: yes , elementary particles rotate in a mathematically convoluted probabilistic way.

In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei. extends to composite particles and also to atoms

So the argument above holds for atoms too. That is why one has nuclear magnetic resonance phenomenon.

All isotopes that contain an odd number of protons and/or neutrons (see Isotope) have an intrinsic nuclear magnetic moment and angular momentum, in other words a nonzero nuclear spin, while all nuclides with even numbers of both have a total spin of zero.

To address this:

Is it that anything considered as a point particle cannot rotate about its own axis? If so, why is that the case?

Strictly, if one considers the center of mass of the particle as a point, and the particle has a volume about that point, there will be a possibility of acquiring angular momentum when, if it is a dust particle, it hits another dust particle; this will give a linear momentum and can (if not head on) also induce rotation about an axis that goes through the center of mass. But if, for simplicity of mathematics to be derived, it is really considered as a particle with no extension about the center of mass  point, there is no way an impact can transfer angular momentum, by construction of the model.
A: A point has a location but zero size aka a radius/diameter/dimension of zero.
In order for something to rotate, some part of the thing must orbit around some other part of that thing. But if something has zero size, that means the entirety of the object exists in the same infinitesmal location in space. So how can multiple parts of an object orbit each other when all those parts are located at the same infinitesmal point in space?
A: As requested by Toba, I would like to summarise my thoughts on the moment of inertia of a point particle and how it leads to a trivial consideration of the energetics of the rotation of a point particle.
This consideration is purely classical.
Moment of Inertia:
The moment of inertia is defined as follows: $$I=\int_V r_{\perp}^2\rho(\vec r)$$
The integral over a particle density $\rho(\vec r)$ multiplied with the square of the distance from the rotational axis $r_\perp$.
The moment of inertia is related to the rotational energy of the object described by the density rotation with an angular velocity $\omega$ as follows: $$E_{\text{rot}}=\frac{1}{2}I\omega^2$$
And also related to the net angular momentum of the system via: $$L=\frac{I}{\omega}$$
Moment of inertia of a point particle:
Now let's consider a point particle, that means we consider a particle density that is peaked at a single point $\rho(\vec r)=m \delta(\vec r)$ (That means it is described by a delta distribution, coordinate frame is chosen in the way that the particle is located in the origin). $m$ is the mass of the point particle. We can now use this definition of the moment of inertia to calculate it for the point particle. $$I_{\text{point particle}}=\int_V r_\perp^2\rho(\vec r)=m \int_V r_\perp^2 \delta(\vec r)=0$$ As the delta distribution gives a non-zero contribution only at $\vec r = 0$ where $r_\perp=0$.
Interpretation:
As we see the moment of inertia of a point particle is always zero, therefore, a point particle never has any intrinsic rotational energy or intrinsic angular momentum. So the internal rotational DOF of a point particle do not contribute in the statistical mechanical as they do not change the energy of the particles.
Angular velocity
It does also not make much sense to talk about the angular velocity of the particle because it because the angular velocity is defined by the angle the object rotates around the rotational axis in a certain time. If the particle is only present at the rotational axis the common definitions of the angular velocity fail:
$$\vec \omega = \frac{r\times v}{r^2}$$
As it describes the rotational between points which is no longer possible in a point particle.
Conclusion

*

*Rotational DOF are irrelevant for point particles as $E_{\text{rot}}=0$

*It does also not make much sense to talk of the rotation of a point particle because in order to define rotation you need two distinct points in the object.

What changes towards the two atomic gas
In the two-atomic gas (two particles of mass $m$, and distance $2\vec d$ perpendicular to the rotation axis, rotation axis between the particles) we have the following particle density \rho = m\delta(\vec r + \vec d) \delta(\vec r - \vec d) therefore the moment of inertia of this system is:
$$I_{\text{2 atomic gas}}=\int_V r_\perp^2\rho(\vec r)=m \int_V r_\perp^2 \delta(\vec r + \vec d)\delta(\vec r - \vec d)=m|d|^2+m|d|^2 = 2 m|d|^2$$
So we have a non zero moment of inertia.
Also the angular velocity $$\vec \omega = \frac{r\times v}{r^2}$$ can be defined and calculated for the system. Therefore the rotation of those particles with respect to the rotational axis matters and leads to a change of the rotational energy of the gas constituent and has to be included in the thermodynamic considerations
Some Links

*

*https://en.wikipedia.org/wiki/Moment_of_inertia

*https://en.wikipedia.org/wiki/Angular_velocity

*https://en.wikipedia.org/wiki/Rotational_energy

*https://en.wikipedia.org/wiki/Dirac_delta_function
A: It doesn't make sense for a point particle to rotate classically. In classical mechanics, a point particle is characterized by it's position and momentum (and mass and charge). The smallest possible rotating object is then a set of two particles that are attached to each other and separated by some small distance. This object can have angular momentum as long as the component particles have different velocities at a given point in time. A point particle cannot have angular momentum in this sense. The only way this would be possible is if we add angular momentum as an additional fundamental quantity to our description of particles. This is more or less what happens in quantum mechanics.
