Why do Russian jets make a squealing sound?
I am by no means an aficionado of Russian fighter aircraft, but I suspect the sound you speak of is caused by intake airflow.

Since Soviet times, their aircraft have ascribed to a design doctrine of being able to fly from damaged or sub-standard airfield surfaces, and incorporate undercarriage &amp; engine intake designs that assume constant presence of engine-killing FOD. They have either a meshed grating, or even an impervious retractable door flap that completely covers the intakes, while having additional doors open on top of the engine intake area to compensate, for ground operations. 

At the 3.10 mark in the video, you hear what is probably the No1 engine spooling up, at about 3.35 you hear the No2 (closest to camera) starting to spool up - within a few seconds you will see a series of small doors above the red number "70" open. This is where the engines are drawing their intake air, away from ingestion of gravel, discarded hardware, small animals &amp; the closely-milling, unconcerned ground crew. 

I suspect that the restriction/turbulence caused by this slightly torturous air path is causing that high-pitched sound. It could also possibly be compressor bleed air being dumped as a normal part of low engine revs/idling. 

At the 9.00 point in the video, you can see the flight intakes completely shut, with the ground intakes open, as it taxies in. Eventually, you even see a ground crewman appear to actually stand in front of the flight intakes while at least one engine is still running. 

There is some discussion &amp; a photo on this page

Would it be viable to install a screen in front of jet intakes to prevent them from sucking in birds?

This aspect of airframe design is a common subject of comparisons between US &amp; Soviet/Russian design doctrine.