Re: spackle came through eggshell paint
Yes, you should have spot primed, then full coat primed the entire wall, then two coats of your top coat.
The reason you're getting these "shiners" is that the spackle is porous and absorbed the paint, while the non-spackled areas the paint stayed on the surface. Spot priming would have sealed these areas, the full prime would have blended the whole wall together, then the two top coats would have given you a perfect finish.
Since you've already painted the entire wall, go back and recoat the shiners with your top coat, you may need to do this twice to get the spots to blend in. Once that is dry, if you now have "reverse" shiners, repainting the entire wall should take care of the problem.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!