Yeah few things eat box elder bugs, they apparently taste and smell something awful...
On that subject if you want a pretty safe and carefree way to control box elder bugs get a pump sprayer, load it with warm water and put a squirt of Dawn dish soap in the water, don't go crazy as it doesn't take much Dawn to do the job... That's it, spray down the box elders as they swarm on the walls or trees, spray in the cracks they are using for entry... The Dawn dish soap and water is like a nuclear toxin to them and it's pretty much harmless to everything else...
As for flies, depending on their mood my chickens will chase flies around all day or ignore them...