Why do we assume that the particles in the gas are indistinguishable? In QM, a set of N particles are indistinguishable only if their combined wave function is either symmetric (bosons) or antisymmetric (fermions) under interchange of two particles. Why do we make this assumption for the combined wave function of the particles in the gas (whose single particle wavefunctions are given by the solutions to the particle in a 3D box problem, as usual)?
I think you have this backwards. We get the result that particles are either fermions or bosons precisely because we assume they are indistinguishable. When you assume particles are indistinguishable in statistical mechanics, you get either fermionic or bosonic models.
Why we assume particles are indistinguishable? Because "indistinguishable" is just a way to say that we are considering all of the characteristics of the particles in our model, that is to say, there is nothing I can measure that I'm not already considering as a variable that would allow me to distinguish two particles.
Example: suppose I have two particles of the same mass and no charge. They have 2 unique properties that identify them: their position and their momentum. They are "indistinguishable" in the sense that if I take particle $1$ and change it so that its position and its momentum mach those of particle $2$, and vice versa, then effectively in the model particle $1$ has become particle $2$, and vice versa, and none of the physics has changed. The system will behave exactly in the same way as if I hadn't done anything. If the particles are identical in everything except position and momentum, how else are you going to label them if not "the particle that's here and slow and the particle that's there and fast"? Hence, it makes no sense to assign labels such as "particle $1$ and particle $2$", but all the labels we need are in the states.
Where the indistinguishabilty assumptions become interesting is that usually we do have these labels for particles: they are the labels of the corresponding Hilbert spaces forming the combined space $ \mathcal H_1\otimes \mathcal H_2$. The process of getting bosons and fermions from distinguishable particles is the process of selecting the portions of $\mathcal H_1\otimes \mathcal H_2$ that is invariant under the swapping of these labels, which is a mathematical way of saying "if I assign to particle $1$ all the properties of particle $2$ and vice versa, nothing changes".
This drastically changes if you artificially make the particles distinguishable, for example, you assign then a fixed position, e.g. "the particle on the left and the particle on the right", and you want to model the behaviour of some other variable, say their spins. Then the system is not invariant under the exchange of the particles, i.e. you want "up down" to be a physically distinct state from "down up", and hence you don't have fermions or bosons anymore. This is the case for example in lattice spin systems.