In the Toyota service manual it talks about a Lithium soap glycol grease being used on the sliding pins of the caliper as well as the rubber seal around the brake caliper piston. A lot of people are saying to use silicone but if Toyota wanted that, they would have specified that.
Silicone can cause swelling to certain rubber parts, and using silicone will cause your brakes to become "sticky". Plus once rubber gets silicone on them it penetrates the rubber and the swelling can become permanent. Sometimes you get lucky and wiping it off is all you need to do. You'll be able to see the swelling with your own eyes if you DIY the brake service yourself.
I've found that this Toyota Lithium soap grease only lasts about a year and needs to be re-applied. But most calipers should be serviced/lubed once a year anyway. If you don't lube this up the caliper WILL seize up and then it's even more $$$ to replace it.
I thought this post was here on this forum but I'm making this a sticky..
The part you need is Toyota part number 08887-01206, called rubber grease 100grams. It's about $13-15 and any Toyota dealer can order it. It's a pink colored grease.