Unlike Haxe, Rust's major syntactic decisions have been openly discussed and archived. For example, the initial type parameter syntax was
[T] (archaea) which then changed to
<T> (or, in some places,
::<T> to avoid ambiguity) to make it familiar to C++ programmers.
I see Haxe itself is largely inspired from ActionScript 3, which is in turn inspired by (now-obsolete) ECMAScript 4. This explains why the final forms of both Haxe and Rust look somewhat similar to each other: both are designed to appeal to users of other curly-brace languages (ES vs. C++) and there are some shared challenges requiring the similar syntaxes (macros, for example). Unlike @Manishearth, I don't think that this is a mere coincidence, but I too agree that this is not really an "influence".