Natural attacks are separate weapons: they use different stats from unarmed strike (including damage die, which monk improves for unarmed strike only), and they are not monk weapons (i.e. they cannot be used for a flurry).
In fact, natural weapons use rather different rules from manufactured weapons and unarmed strikes. They do not use iteratives or two-weapon fighting; instead, when you full-attack, you simply get one attack, period, with each natural weapon you have. You can combine manufactured weapon attacks (which, for these purposes, the unarmed strike effectively counts as) with natural weapon attacks. In this case, you attack however many times you would normally (iteratives from high BAB, extra attacks from two-weapon fighting or haste, etc.), and then also make any natural weapon attacks you still can, but you treat all of them as secondary (i.e. −5 attack penalty and only ½×Str to damage). Any natural attack associated with a limb used to wield the manufactured weapon(s) you already attacked with (e.g. a claw attack on your sword hand) cannot be used in this full-attack.
The interesting bit here is that flurry of blows is a full-attack. Flurry of blows is a class feature that offers an option to increase your number of attacks in exchange for an attack penalty (much like two-weapon fighting), and it specifies that “[t]hese attacks can be any combination of unarmed strikes and attacks with a monk special weapon.” Emphasis mine: the attacks from flurry of blows must be with monk weapons. But that does not stop you from also including natural weapon attacks, as secondary attacks, as part of the flurry of blows.
Finally, Feral Combat Training allows a chosen natural weapon to count as unarmed strike for feats that require Improved Unarmed Strike. This does not include the monk’s improved unarmed strike damage. A special benefit for monks does allow the natural weapon to be used as the weapon in a flurry of blows. One assumes that this is instead of using it as a secondary natural attack (in effect, that limb has already been “used”), so this is only worthwhile if your natural weapon is really good—you effectively lose an attack when you use this feat.