Will Brown Areas After BBL Flake Off?

Those spots will fall off on their own, revealing less spotty skin beneath.  You will be happy with this.

No need for alarm, sometimes a mild exfoliating cleanser can help speed the removal of these spots a little.

Stay out of the sun while you are healing, use sunscreen and wait for the younger looking skin to emerge!

Coffee ground pigment is a common observation after IPL or BBL on the skin. This is a good sign and will resolve in approximately 14 days. The purple areas likely represent purpura (small bruise) where a blood vessel (telangiectasia) was also treated by the laser. I would expect this to fade over 14 days as well. Make sure to avoid sun and wear sunscreen if your are out and about. 
Matthew Muellenhoff, DO, FAOCD

The dark spots are the sun damaged areas both seen and unseen by you prior to treatment.  If you doctor provided post-treatment instructions, make sure you refer to them to know what to expect and to ensure you are following after care procedures.  Your external barrier to the elements including sun has been altered and more susceptible to damage now so use UV protection heavily and daily.  You should see overall lightening of your hyperpigmentation over the next 30 days.

What you're describing sounds perfect. It sounds like you've had a good response to the treatment, with the spots turning darker and looking patchy.  You should expect these spots to flake off approximately one week to 10 days after treatment.  I think you will be quite pleased with the results.  Though you will see marked improvement with one session, typically, three sessions,
spaced two weeks apart are required for most effective removal of brown spots. Enjoy you're new, more even skin!

Darkening of brown spots such as freckles and lentigos followed by flaking off over a period of 7-14 days is a normal response. Make sure to be wearing sunscreen while your skin is healing and gently washing with a non-scrub cleanser.