It sounds like you have the liquid BKF if you had to shake it first. I don't know how it compares to regular BKF, but I got a new countertop out of having my fabricator use BKF while cleaning up after installation.
The reason I always caution against it is that it does dull the finish. While YOU may not see it on your counter and recommend it, someone else may use it on their counter and end up with a permanent etch. The biggest factor is the backlighting from a window or a light fixture that throws light on your counter. That's where you'll see the glossiest sheen of a counter, and that's where you will most likely see an etch. Etches are not removable from quartz like they are with granite. The dull spots from BKF on my peninsula were visible in the light from a sliding glass door.
I would never use anything other than what the manufacturer recommends. If there's bleach in your cleaning product, it can yellow the resins over time. What if you recommended that to someone who owns a pure white quartz countertop? You shouldn't need anything more than a soapy cloth or sponge to clean quartz. I use Method All-Purpose Surface Cleaner on occasion. You shouldn't get stains on your quartz either. I haven't had one in six years.
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Do not use Barkeepers friend on quartz....dulls and etches over time,
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