Defining binary variables on box constraints in mixed integer linear/convex program

Question

I have $n$ variables $y_1,\dots,y_n\in\mathbb R$ with no upper bound and no lower bound.

I want to define a binary variable $b\in\{0,1\}$ on condition that $b=1\iff \wedge_{i=1}^ny_i\in[0,1]$.

How do I do this in Mixed Integer Linear Programming?

I want to do this with $O(\log\log n)$ additional integer variables at most and perhaps using $poly(n)$ convex constraints.