If basic symmetry and homogeneity assumptions about the Universe hold, then yes, all massless real particles (see Anna V's answer for virtual particles must travel at a universal constant $c$, the speed of a massless particle, in all frames of reference.
Given these basic symmetry and homogeneity assumptions, one can derive the possible co-ordinate transformations for the relativity of inertial frames: see the section "From Group Postulates" on the Wikipedia Page "Lorentz Transformation". (Also see my summary here). Galilean relativity is consistent with these assumptions, but not uniquely so: the other possibility is that there is some speed $c$ characterizing relativity such that $c$ is the same when measured from all frames of reference. Time dilation, Lorentz-Fitzgerald contraction and the impossibility of accelerating a massive particle to $c$ are all simple consequences of these other possible relativities.
So now it becomes an experimental question as to which relativity holds: Galilean or Lorentz transformation? And the experiment is answered by testing how speeds transform between inertial frames. Otherwise put, the experimental question is are there any speeds that are the same for all inertial observers?. The question is not about measuring the values of any speed, but rather, how they transform. Now of course we know the answer: the Michelson Morley experiment found such a speed, the speed of light. So there are two conclusions here: (1) Relativity of inertial frames is Lorentzian, not Galilean (which can be thought of as a Lorentz transformation with infinite $c$) and (2) light is a massless particle, because light is observed to go at this speed that transforms in this special way.
Notice that at the outset of this argument we mention nothing about particles or any particular physical phenomenon (even though special relativity's historical roots were in light). It follows that, if $c$ is experimentally observed to be finite (i.e. Galilean relativity does not hold), then the specially invariant speed is unique: it can only be reached by massless particles and there can't be more than one such $c$ - the Lorentz laws are what they are and are the only ones consistent with our initial symmetry and homogeneity assumptions. So if we observed two different speeds transforming like $c$, this would falsify our basic symmetry and homogeneity assumptions about the World. No experiment gives us grounds for doing that.
This is why all massless particles have the same speed $c$.
Update: Experimental Results
As is now common knowledge, the gravitational wave event GW170817 and gamma ray burst GRB170817A give strong experimental evidence of the equality of the speeds of light and gravitation. As discussed in:
Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A
the 1.7 second time delay between the gravitational wave arrival and the gamma ray burst, together with conservative assumptions about other sources of delay, yields an experimental bound on the fractional difference between the speed of light and of gravitation:
$$\frac{v_g-v_{em}}{c} \leq 3\times 10^{-15}$$
an impressive experimental bound indeed. Within the next 10 years, we probably shall see several such events, and thus this experimental bound will tighten further (unless something really theoretically unforeseen happens!).
Mass From Confined Massless Particles
Incidentally, if we confine massless particles, e.g. put light into a perfectly reflecting box, the box's inertia increases by $E/c^2$, where $E$ is the energy content. This is the mechanism for most of your body's mass: massless gluons are confined and are accelerating backwards and forwards all the time, so they have inertia just as the confined light in a box did. Likewise, an electron can be thought of as comprising two massless particles, tethered together by a coupling term that is the mass of the electron. The Dirac and Maxwell equations can be written in the same form: the left and right hand circularly polarized components of light are uncoupled and therefore travel at $c$, but the massless left and right hand circular components of the electron are tethered together. This begets the phenomenon of the Zitterbewegung - whereby an electron can be construed as observable at any instant in time as traveling at $c$, but it swiftly oscillates back and forth between left and right hand states and is thus confined in one place. Therefore it takes on mass, just as the "tethered" light in the box does.
Simply put, the Higgs isn't charged under the strong force. It doesn't have standard electrical charge either. The $W^{\pm},Z$ bosons aquire a mass through the Higgs mechanism because the Higgs itself is charged under the weak force. Leptons aquire masses through the Higgs mechanism because they too interact with the Higgs.
No Higgs interaction means no effective mass.
You ask why the Higgs does not interact with gluons. It has to do with the quantum numbers (charges) of the fundamental particles in the standard model. It turns out that you aren't allowed to freely choose quantum numbers for the different particles. If you made a bad choice, you'd violate gauge invariance and have an inconsistent theory. This puts relatively strict constraints on the allowed quantum numbers. check out Gauge Anomaly for more details.
Basically, the known quantum numbers of the other standard model particles constrains the allowed quantum numbers of the Higgs, specifically prohibiting a gluon-higgs interaction. if you wanted to add that interaction, you would necessarily imply the existence of other particles in order to balances all the charges of the theory. I don't know if that would be possible, but its a matter of simple algebra to figure it out.
Best Answer
Because in special relativity and in terms of conserved momentum and energy velocity is given by:$$\vec{v} = \frac{\vec{p}c^2}{E},$$ and energy and momentum are related by:$$E^2 = \left(mc^2\right)^2 + (pc)^2,$$ giving: $$\vec{v} = \frac{\vec{p}}{\sqrt{\left(mc\right)^2 + \left(\vec{p}\right)^2}}c.$$ There is no real momentum, $\vec{p}$ that can give a velocity $\vec{v}$ higher than $c$, and only infinite momentum gives $v=c$ when $m\neq0$.
For the specific question you have, the individual gluons cannot accelerate, but the frequency of gluons traveling in one direction can be different than the frequency of the gluons traveling in the other. This gives the box of gluons, on average, a momentum and kinetic energy. The inertia of the box is a property it has, again averaged over a time long enough for the gluons to bounce back and forth, that is caused by the way gluons reflecting from a moving wall will change frequency in a way that exerts a force on the wall and changes the frequency of the gluons.
It's a classic undergraduate physics problem to calculate the change in frequency caused by the Doppler effect of light reflecting off of a moving object. The momentum and energy needed to cause that change in frequency have to come from somewhere, and it comes from the force exerted on the object the light reflects from.