You cannot overload assignment. Moving a variable from one location to another is a core component of Rust's ownership semantics and is not overridable.
Another answer suggests that you custom-implement the Copy trait. This makes no sense, as there's nothing to implement:
pub trait Copy: Clone { }
You could implement Clone for a type, but to use clone you have to call it explicitly:
let foo = bar.clone();
The actual assignment is still just copying bits from the right-hand side to the left-hand side, the only difference is that you don't give up ownership of bar.
If your type can be duplicated by simply copying bits, then it's appropriate to implement Copy. If it can be duplicated by executing some kind of function, then it's appropriate to implement Clone. There is no way I know of to implicitly execute code at any given assignment of a type (and I count that as a good thing).