I am fixing up an older Schwinn Protocol 1.0 for a friend and I need to replace the mechanical, post mount disc brake caliper. (I took apart the old one and it is missing a magnet and in spite of cleaning, sticks in parts of the travel.) What can I replace it with? There is no ID on the caliper. Is any post mount mechanical disc brake caliper OK? Are front and rear disc calipers different?
That is a Post Mount caliper, which is bolted to an International Standard (IS) mount adapter.
Upshot - you need to keep the adapter, mounting bolts, and spherical washer sets when you fit your new brake caliper.
I went through something similar with an Avid BB5 mechanical caliper that I simply couldn't tune. Even new pads didn't help. In the end I bought the lowest-spec Shimano hydraulic brake kit and it bolted up perfectly. Shimano BL-MT401 + BR-MT410 came pre-bled too with enough hose to fit perfectly.
Caliper alignment was simple. The hardest part was stripping the bars to get the brake lever to slide up where it was needed.
Do note that pre-bled hydraulics mean you need to pass the caliper or the lever through any holes needed. For most bikes its no big deal, but if you have internal cable routing, well you won't be able to do that and will have to strap the hose to the outside of the frame.
In short - you can use any Postmount mechanical caliper, or take the opportunity to upgrade to hydraulic for not a lot more cost, if any.
Whether the front and rear are interchangeable depends on the brand and model. If they don't list a front and a rear it will work in either position. Mechanical disc are now available on even entry level bike so the price of replacements are reasonable. The caliper in your image looks like an Avid BB3 or an off brand knock-off. Something to consider is while the calipers may look the same there is a difference between Road and MTB calipers. This has to do with the pull ratio of the levers. Cheap calipers tend to come with cheap pads. Cheap pads are more likely to squeal and experience fade. While you may not need the best you want something dependable.