See this video. It seems that Boys of Silence can be killed!
The steps outlined in the video are as follows.
- Equip the Spectral Sidekick Gear.
- Find a spare weapon.
- Drop/pick up the spare weapon until the gun has a clear line of sight to the Boy of Silence.
- Get the Boy of Silence to turn towards you with a yellow light.
- Take cover and let your "ghostly ally" hit the Boy of Silence.
- Repeat step 4-5 until the Boy of Silence dies; if the effects of Spectral Sidekick end before the Boy of Silence dies, go back to step 3.
- Do an Irish jig over his dead body.
- The description for this Gear is "Dropping a weapon creates a ghostly ally for a few seconds".
- In the video linked to above, which shows a part the level called "The Atrium", there is a weapon lying in front of some crates on the left, in the room right after that with all the Motorized-Patriot masks, just before you encounter the first Boy of Silence in the game.
Edit: here is an alternative method applicable to (SPOILER)...
the Boy of Silence that sneaks up on you just before you rescue Elizabeth from her tormentors, near the end of the game.
Just one-shot the Boy of Silence (in the mouth, if that matters) with the shotgun, as in this video; which difficulty level is used in the video is unknown, though.