How do I get both rear brake calipers to stop sticking?

A brake caliper can stick because the caliper slide pins are corroded, the anti-rattle clips are corroded, or the caliper pistons are sticking (replace calipers).

To add to user9181, when changing brake pads, its important to properly lubricate the moving parts. If you are doing this yourself, the new pads/clips should come with a little diagram showing where to apply lubricant to the clips. If you have a shop do the work, you can only hope they do all this. If the caliper pistons are stuck, it is possible to rebuild these (basically, installing fresh rubber boots and cleaning up the sliding surfaces), but a new set of calipers is fairly cheap. Good luck!