The point at which boundary conditions are specified is subtle. You can find this point by adding a constant vector to $\vec{E}$ in each step until it varies the equation: it should not change any equation until the boundary is specified. Since the flux of a vector field is unaffected by an added vector constant, **the boundary condition was first specified when $\vec{E}$ was assumed to be spherically symmetric everywhere**, which caused it to fall out of the integral. 

**In what way is that a boundary condition?**

It carries extra information that Maxwell's Equations alone do not: it assumes that the universe is rotationally invariant. The boundary condition was that $\vec{E}(r, \theta, \phi) = \vec{E}(r, \theta+\theta_0, \phi+\phi_0)$ for any $\theta_0, \phi_0$, if putting it into an equation makes it seem more legitimate.