The integer overflow issue is only part of the problem. There are a couple underlying issues that make this exploit feasible: certain regexes produce extremely long bytecode, the integer overflow issue, and finally, a double free.

Rust will not protect you against exploding regexes, it will not protect you against integer over/underflows (unless you specifically call safe wrapping methods), but it will protect you against double frees if you use it sensibly (nothing prevents you from breaking the borrow checker and messing up memory if you write an unsafe {} block).