Yes, you need those springs, or the pads will vibrate around terribly. You should also have two wire springs that hook into the edges of the pads to spread them apart. There are little holes in the pad edges for this. If they're missing it will also contribute to a squeal.
If the rotors were changed with the pads there's no reason to turn them. I'd avoid ever turning/cutting a rotor. If it's worn unevenly replace it.
The biggest causes I see for brake squeals are:
1. Sticking slide pins. They cause the pads to drag. Greasing the pins sometimes isn't enough to make them slide freely. Remove the rubber bushing from the caliper body and use a round file to clean the rust out of the hole, then paint the caliper there to prevent return of rust. Then put the rubber bushing back in and grease the pins. Should slide much easier.
2. Pads not sliding freely in the caliper bracket. Aftermarket pads these days are garbage, for the most part. I install about 15 sets of brake pads a week, and about 10 of those sets have pad backing plates that are stamped poorly and don't fit in the caliper right. If they don't slide freely in the caliper, they will drag and squeal. Get a file and clean up the contact points so they fit the calipers with about .020" gap for heat expansion and future rust swelling. Paint the edges to prevent rust when you're done filing.
3. Sticking caliper pistons (or bad brake hoses). If the caliper pistons didn't push back easily when you did the brakes, the calipers might be bad. Sticking calipers will cause squeal. Open the brake bleeder and see if the caliper is easier to push back. If it's not then the caliper is bad. If it is easier, your brake hose is bad on that wheel.
4. Missing anti-rattle/anti-squeal hardware. You already said this was missing, so fix it.