Bears still have heads. Headbutts are unarmed strikes.
That's not how Martial Arts works in 5E. Check the rule:
"You gain the following benefits while you are unarmed or wielding only monk weapons and you aren't wearing armor or wielding a shield." (SRD 5.0, p.26)
In previous editions, there was an idea that a monk could make an unarmed strike with anything -- a head, a leg, whatever -- but that doesn't apply in 5E. Instead, you look at the brown bear, for instance, and note that both its bite attack and claw attack are defined as 'melee weapon attacks'. (SRD 5.0, p.364) While an unarmed strike is also considered a 'melee weapon attack', that doesn't mean that all melee weapon attacks are equivalent to unarmed strikes -- a DM could absolutely rule that a beast, all of whose attacks are considered melee weapon attacks, is not unarmed for the purpose of the Martial Arts rule. In fact, this is supported in the SRD -- "The most common attacks that a monster will take in combat are melee and ranged attacks. These can be spell attacks or weapon attacks, where the 'weapon' might be a manufactured item
or a natural weapon, such as a claw or tail spike." (SRD 5.0, p.262, emphasis mine)
Flurry of Blows doesn't require the use of a Monk weapon, so you can Attack with claws and then Flurry.
But you need to be able to make unarmed strikes to get the benefit of Flurry of Blows -- if you don't have unarmed strikes, because you're a bear with a bunch of natural weapons, then you don't have the ability to make any attacks via Flurry of Blows, because you're not unarmed. It doesn't matter that the bear has a head; that it has claws that are considered natural weapons (that aren't monk weapons) is enough to disqualify it from using unarmed strikes.
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Pauper