The air is coming sufficiently fast that all you could hope for from a “screen” (it’d have to be something very solid and substantial to withstand the thrust) is that it would dice up the bird before the carcass reached the fins. But even then you would still have the same mass of bird in the engine.
Also, I’m sure such screens would significantly reduce the efficiency of the engines. (Basically they’d act like baffles). If they were far enough out to be thin screens, they would cause an enormous amount of drag.
The main reason they wouldn’t do such a thing is that bird strikes are so uncommon, and most of them are survivable. What made the Hudson plane so unusual is that birds struck both engines at the same time while the plane was still too low to do a restart. How many times has that happened in history? I’m pretty sure the cost/benefit in this case would be prohibitive.