This is only a partial answer but maybe it can lead you somewhere. We can define the join of two cubical sets as follows:

If $C$ and $D$ are cubical sets, then $(C \wedge D)_n$ is the set $C_n \sqcup \bigsqcup_{n = i+j+1} (C_i \times D_j) \sqcup D_n$. For $x \in C_i$ and $y \in D_j$, the faces of the cell $x \wedge y \in (C \wedge D)_{i+j+1}$ are given by:

- $\partial_k^\alpha (x \wedge y) = (\partial_k^\alpha x) \wedge y $ for $1 \leq k \leq i$
- $\partial_k^\alpha (x \wedge y) = \epsilon_{i+1}^j x$ for $k = i+1$ and $\alpha = -$ (where $\epsilon_{i+1}^j$ is $\epsilon_{i+1}$ applied $j$ times).
- $\partial_k^\alpha (x \wedge y) = \epsilon_1^i y$ for $k = i+1$ and $\alpha = +$
- $\partial_k^\alpha (x \wedge y) = x \wedge \partial^\alpha_{k-i-1} y$ for $i+2 \leq k \leq i+j+1$.

I haven't checked, but I expect that one should be able to find suitable formulas for the degeneracies (and connections if $C$ and $D$ come equipped with connections) so as to make the join into a bifunctor. However one has to be careful: this bifunctor does *not* define a monoidal structure on cubical sets (even with connections): it is not associative.

Anyway, if you start from the terminal cubical set $\top$, then $\top^{\wedge n}$ (warning: you have to choose a suitable bracketing for this to make sense: for example $(\top \wedge \top) \wedge \top$ and $\top \wedge (\top \wedge \top)$ are not isomorphic) is a good candidate for the image of $\Delta[n]$. I do not know however whether all the morphisms in $\Delta$ have an image between the $\top^{\wedge n}$.

A word on the join of cubical sets: AFAIK it appears nowhere in the literature. It doesn't have very good properties on cubical sets. On cubical categories however it should be a monoidal product, and should correspond to the join of $\infty$-categories defined by Ara and Maltsiniotis in "Joint et tranches pour les $\infty$-catégories strictes"

Alternatively there is a morphism from $\Delta$ to $\square_c$ that sends $[n]$ to $[n+1]$, degeneracies to connections $\Gamma^+$ and faces to faces $\partial^+$. This is just a way to say that in $\square_c$, $[1]$ is a monoid object. General considerations on Kahn extensions tell you that it induces a left-adjoin from simplicial sets to cubical sets with connections. I believe that is the same as teh construction given by @KarolSzumilo.

There is also a dual construction obtained by considering the other monoidal structure on $[1]$, using $\Gamma^-$ and $\partial^-$. Finally this construction can be adapted if you work between *augmented* simplicial sets and cubical sets with connections.