No!
Best not to use insecticides preventatively, it can cause resistance....
....then they might not work if and when you really need them.
Keeping wild birds and rodents out of your coop and run is the best preventative.
..and do a bug check regularly.
Definitely looks like a quill....and is....kinda.
Usually called Pin Feathers.
New feathers growing in can be itchy business,
lots of preening or scratching where the beak won't reach.
First make sure they have bugs....and what kind.
7 week old birds may be going thru a juvenile molt.
The oil gland is at the base of tail, they use that when preening.
Have you checked them over real well for mites and/or lice?
Google images of lice/mites and their eggs before the inspection...