Yep, they will. And spreading seed under the cedars, as long as the branches are low to the ground, will help a lot. The birds are used to the dangers of hawks--they'll cope.
As for the squirrels and rabbits and mice, they are in no danger from this particular hawk (hawks tend to specialize in mammals or birds, but usually not both). Our sharp-shinned and Cooper's hawks clear the birds out in a flash--the squirrels, rabbits and voles don't give a hoot. If a red-tailed hawk goes over, it's the opposite--the little birds keep eating, the mammals go running...
I just found evidence of a hawk strike on a mourning dove out by the feeders. Lots of feathers and no birds to be seen. Must've just happened. I saw a Cooper's the other day as it missed a strike on a different dove. Usually, since we have ample cover and the doves are obviously on the alert (despite one having been taken, they're very flighty and cautious today), we leave the feeders up. Birds feeling threatened will be scarce at the feeders for the next few days and the hawk will find less prey, so will move on by itself.