Reading through the unchained rules I didn't find a mentioning anywhere so is an unchained Monk able to use flurry of blows in combination with claws?
By default, monks cannot include natural attack in a Flurry of Blows. The only way to do so is by taking the feat Feral Combat Training. The feat is limited to one specific weapon, so you can choose one of your claws.
Choose one of your natural weapons... If you are a monk, you can use the selected natural weapon with your flurry of blows class feature.
Otherwise, the answer is no. It is true that the unchained version of Flurry of Blows is worded differently from the core version, which explicitly forbids the use of natural weapons in this way. However, the unchained monk's ability still has similar limitations:
When using this ability, the monk can make these attacks with any combination of his unarmed strikes and weapons that have the monk special weapon quality.
I don't know if it was an intentional design choice to not mention natural attacks in the unchained rules. Regardless, natural attacks are not unarmed attacks, and natural weapons are not typically able to qualify as a "monk" weapon.
As an alternative to Flurry of Blows, a monk can full attack with all of their iterative attacks (from BAB) and their natural attacks (at a penalty). Like the flurry, doing this is a full attack action, so the monk can't perform multiple full-attack actions in one turn. As explained by the game's designers, once a monk performs a Flurry of Blows, the full attack ends, so they cannot make additional attacks with natural weapons that turn.
Yes and no.
The flurry of blows attacks are restricted to “monk” weapons and unarmed strikes, and claw natural weapons are not that (unless you have the Feral Combat Training feat). So your flurry of blows attacks have to be made with some other weapon.
However, a flurry of blows is still a full attack, it just gets some extra attacks. And it is allowed to combine natural weapon attacks with your regular weapon attacks (i.e. those made with manufactured weapons or unarmed strikes) during the same full attack—and flurry of blows is a full attack. So after your flurry of blows, you can also attack once each with your natural weapons. They are always treated as secondary attacks when used in this fashion, even if they would usually be primary.
Relevant rules text:
At 1st level, a monk can make a flurry of blows as a full-attack action.
Creatures with natural attacks and attacks made with weapons can use both as part of a full attack action (although often a creature must forgo one natural attack for each weapon clutched in that limb, be it a claw, tentacle, or slam). Such creatures attack with their weapons normally but treat all of their available natural attacks as secondary attacks during that attack, regardless of the attack’s original type.
So with these rules, you can use natural attacks “alongside” your flurry of blows. Feral Combat Training lets you use the natural attacks for the flurry of blows attacks too.
Note that this rules interaction is really confusing for a lot of people. The question has been asked and answered over and over. It is something you want to discuss with your GM. It’s even quite plausible that Paizo themselves don’t intend it (though to my knowledge they have never addressed it).
Ultimately, though, natural attacks are not strong, and monks, even unchained monks, are really not that strong. You need all the help you can get. Feral Combat Training should never have been written; natural-attack monks should have just been a default option.