Yes. 

The problem's statement can be reformulated as follows. Suppose that for any set $X$ there exists a function $f: (2^X\setminus \{\emptyset\})\to X$ mapping any nonempty subset (of escapees) to one of its elements (which guarantees that a participant will refuse to participate in the escape). Is it equivalent to the Axiom Of Choice?

The Axiom of Choice states that for any set $X$ of nonempty sets $Y_i\in X$ there exists a function mapping any set $Y_i$ to an element of the set. 

But the equivalence is fairly trivial. Indeed, the AOC follows by constructing the function $f$ for $\cup Y_i$ and applying it only to the sets $Y_i,$ while the function function $f: (2^X\setminus \{\emptyset\})\to X$ mapping any subset to its element is created by applying the AOC to all nonempty subsets of $X.$