I've worked on a lot of brakes over the years, but mostly in US made cars and trucks. But I can answer your BMW question. Most vehicles have hardened steel pins and cast iron calipers with drilled holes for the pins to slide in. They have rubber seals near the head to keep water and dirt out. One of the top causes of brake repairs (not counting worn pads, warped/scored rotors, and ABS/hydraulic problems) is seized slide pins. Since both parts contain iron they will both rust together (although the hardened pins rust more slowly). They will always get moisture in them over time, and dust and grit if the seals are worn. They MUST be greased unless the manufacturer says otherwise. The Manufacturer is always right. Usually. The grease should be a non-petroleum base such as silicone (aka dielectric), since rubber is petroleum based and will absorb certain chemicals in the grease and will soften and swell, which leads to failure.
The reason BMW and a few others do not need grease is because they use a more expensive system of self lubricating bronze bushings and stainless steel pins. Neither material will ever rust. The only thing that will cause them to seize up is if they are greased and the grease attracts and holds grit (especially silica dust from sand, which is harder than most steels) and the grease/grit creates a very solid material as the VOC's evaporate from the grease over time. The same goes for plastic bushings, no grease or oil, they are self lubricating.
I would not use the grease you have on any brakes unless you can find literature that states it is safe for slide pins and rubber parts. I use CRC brand Brake & Caliper Grease which states on the bottle "Will not damage rubber or plastic components, freeze, melt, or wash out." and can be used on all areas on brakes "including caliper slides, bolts, and pins." Most auto parts places carry CRC products and can supply it. I get it from Auto Zone for $16/8 ounces. Be sure and use a thin layer on the pad ears where they slide in the channels. Remember, YouTube is your friend.
The majority of the time that brakes make a loud noise when applied is because the pads and rotors are glazed (a highly polished surface from wear). Often this can be corrected without a brake job. Search online for "Bedding Brakes" for instructions where you get on a deserted straight section of road and reach certain speeds and then brake firmly to lower speeds. It is not difficult and is completely safe to do, just make sure there are no other drivers around who may call 911 and report a possible DUI. Don't ask me how I know this.