When the complex scalar in the vector multiplet has a non-trivial VEV $a$, then indeed the gauge group is spontaneously broken and we have the Higgs mechanism. As any field of the vector multiplet, the complex scalar is adjoint valued and so, classically, the gauge group is broken to the stabilizer of $a$ in the adjoint representation. For generic $a$, it is simply the maximal torus of the gauge group but for special $a$ the unbroken subgroup can be larger. For $G=U(N)$, by gauge transformation one can assume $a$ diagonal, $a=diag(a_1,...,a_N)$. If all the $a_i$ are distinct then $U(N)$ is broken to its maximal subgroup $U(1)^{N}$.

The Young diagrams come from the localization at the fixed points of the instantons moduli space. The Nekrasov partition function is a priori defined as a path integral. Using supersymmetry and after computation of the perturbative part, this path intergral can be reduced to a sum over the instanton number of finite dimensional integrals over the moduli spaces of instantons. These integrals are then computed by localization with respect to the action of the group $T \times U(1)^2$ where $T=U(1)^N$ is the maximal torus of the gauge group $U(N)$ and where $U(1)^2$ is the maximal torus of the group $SO(4)$ of rotations of the Euclidean four dimensional space. $(a_1,...,a_N)$ will appear as the equivariant parameters with respect to $T$ and $(\epsilon_1,\epsilon_2)$ as equivariant parameter with respect to $U(1)^2$. The key point is that that the fixed points of $T \times U(1)^2$ acting on the moduli space of $U(N)$ instantons of instanton charge $k$ are naturally parametrized by the data of $N$ Young diagrams whose sum of sizes is equal to $k$. To understand that, one has to realize a subtle point: the instanton moduli space is in general singular, due to point-like instantons, and to have well-defined integrals, one has to consider a smooth moduli space. A way to resolve the singularity is to turn on a noncommutative deformation of the spacetime and the Young diagrams parametrize fixed points on the moduli space of instantons on this non-commutative spacetime. As we consider fixed points for the action of $U(1)^2$, each fixed point corresponds to a non-commutative instanton localized at the origin of spacetime. As we consider fixed points for the action of $T$, each fixed point corresponds to a non-commutative $U(N)$ instanton which is in fact a direct sum of $N$ non-commutative $U(1)$ instantons. The notion of $U(1)$ instanton may sound stange: in the commutative world, such things do not exist but they exist in the non-commutative world.

So it remains to realize the following: to give a $U(1)^2$-invariant non-commutative $U(1)$ instanton of charge $k$ at the origin of $\mathbb{R}^4$ is the same thing that to give a Young diagram with $k$ boxes. To do that, one has to understand what is a non-commutative $U(1)$ instanton of charge $k$ sitting at the origin. Let me just say that it is a $U(1)$ gauge field configuration with a singular behavior at the origin and that if the instanton is $U(1)^2$ invariant, this singular behavior is completely determined by the shape of a Young diagram. (A mathematically precise formulation of a non-commutative $U(1)$ instanton is as the ideal sheaf of a non-reduced point in $\mathbb{C}^2$. A Young diagram is exactly a way to describe a possibly non-reduced $U(1)^2$-invariant point sitting at the origin of $\mathbb{C}^2=\mathbb{R}^4$).