Would the extra damage die from the Brute subclass still get added to the monk's Martial Arts and Flurry of Blows attacks?
Not with unarmed strikes
As of the first Player's Handbook errata unarmed strikes no longer count as weapons.
Instead of using a weapon to make a melee weapon attack, you can use an un-armed [sic] strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons).
The Brute Damage requires a weapon:
Whenever you hit with a weapon that you’re proficient with and deal damage, the weapon's damage increases
This means that Flurry of Blows can not use Brute Damage as it only uses unarmed strikes (emphasis mine):
Immediately after you take the Attack action on your turn, you can spend 1 ki point to make two unarmed strikes as a bonus action.
You could use a Monk Weapon with Martial Arts
Martial Arts affects certain weapons in addition to unarmed strikes (emphasis mine):
You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.
You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon.
If using a monk weapon, the Brute Damage would apply.
The Brute's Damage is not added to unarmed strikes
The Brute archetype says:
Starting at 3rd level, you’re able to strike with your weapons with especially brutal force. Whenever you hit with a weapon that you’re proficient with and deal damage, the weapon’s damage increases by an amount based on your level in this class, as shown on the Brute Bonus Damage table.
Even though unarmed strike is considered a weapon attack, it is not considered a weapon, so the extra damage from Brute Force would not be added to it as it specifically says "hit with a weapon" rather than "make a melee weapon attack" like most features do, although it would of course still be added to any weapon attacks you make with a weapon.