Are there rubber bushings ON the end of one of the slide pins, usually the lower one ? If so, these bushings are frequently the cause of sticky / seized slide pins. The rubber swells, and starts to bind in the bore. The bushings are to prevent rattling, so they are not absolutely needed. You can simply remove the bushing, or replace the pin with the regular pin. It will slide smoothly, but it might rattle. On one of my cars, the calipers did not rattle at all without the bushing, on another car it drove me crazy and I replaced the bushings.
And careful with the brake lube. I am now using a different brake lube again, one that is supposedly compatible with rubber brake components. The last one was Syl-glide, and the bushings were still swollen. Before that Perma-tex synthetic brake grease ... the purple stuff. I think their new formulation is silicone based, and orange ...