The Brand Squire can be found in the Unknown Arena at the top of Conflux Mountain, in the region of Chersonese. To enter the arena you will need to take the Strange Rusted Sword from the stone at the top of the mountain. Now you will need to combine it with the following items to craft the sword ‘Brand’:
Chemist’s Broken Flask - Acquired from Tuan the Alchemist in Levant, Abrassar
Mage’s Poking Stick - Acquired from Laine the Alchemist in Monsoon, Hallowed Marsh
Blacksmith’s Vintage Hammer - Acquired from Quikiza the Blacksmith in Berg, Enmerkar Forest
Once you have all three items, combine them with the Strange Rusted Sword to craft ‘Brand’ and return to the peak of Conflux Mountain. You will now be able to interact with the stone from which you initially retrieved the sword and enter the Unknown Arena.
Immediately upon entering you will be confronted by a boss named Brand Squire. He can be categorised as a spellblade, with powerful basic melee attacks and two spells. The first spell creates a cloud of red and purple coloured smoke, dealing 5 Decay damage and healing the boss for 50 health. This can be interrupted by chaining a number of heavy and light attacks. The second spell produces a wall of fire which deals 15 Fire damage and inflicts the Burning status effect (3 damage per second for 40 seconds.) You can notice the animation for this when the boss squats to the ground and curls his body. The healing smoke cloud is performed almost without warning, but if you continue to pressure the boss with swift attacks, he will rarely perform it.
The best weapon to bring to this fight is probably the Worldedge Greataxe. See here for how to acquire it. The main advantage of this axe is the presence of Decay damage; the only element which Brand Squire is vulnerable to. In addition, the 48 impact damage will allow you to chain heavy attacks to consistently knock the boss down, preventing the use of both powerful spells. So long as you can avoid the spells by stepping aside or dodging, and continue to apply pressure by chaining heavy and light attacks, you should be able to repeat this process and whittle the health of the boss down within 10 minutes or so.