Well it would run since the only real requirements for Rust is a DirectX 9 compatible card.
But the R7 240 is really more of a if-you-don't-have-a-built-in-gpu-then-take-this, The price/performance of cards like the R7 240 and R7 250 is very bad, i would strongly suggest atleast getting the money for a R7 260 or GT750, that's where you'll see some much better price/performance action.
Of cource if the R7 240 is your only choice and you really need a gpu now, then it would probably run Rust yes, but if you can wait or have the extra money for it, really, get the R7 260 or GT750.
having the more expensive card would be my suggestion, both for the cheapest gpu and for better performance.
1: it will last you longer because of the extra performance, making the time before you'll need to make a costly upgrade again longer, and that makes it less expensive than it's pricetag when looking ahead in time.
2: it will perform better in games AND it will run every game on atleast lowest settings, which i can't say about the R7 240 at all.
This counts for both of my recommended cards. Hope this helps you