It was my understanding that the English and the French were in many wars on opposing sides in the 16th century, so why would King Henry VIII hire a French executioner to kill Anne Boleyn instead of an English executioner?
France and England were in a state of perpetual high and low-key enmity. They weren't warring all the time, and in fact Henry VIII's sister Mary had married the then deceased French king.
The executioner was from an English-occupied territory in France called Calais, so he wasn't quite like hiring someone who lived in France proper.
Burning was a terrible death, and axemen beheadings could be terribly botched. The hiring of the presumably highly esteemed swordsman executioner from Calais was seen as a benevolence and mercy towards Anne. Though considering he'd been hired before Anne had even been convicted in the kangaroo court, it was a small mercy. Henry made Anne wait to know her fate, and she was possibly bribed to agree to the annullment by offer of a quicker, cleaner death.
Using the Calais executioner went towards Henry VIII's attempt to not look like the shady player he was. It wasn't an issue that the swordsman was from Calais, especially as Calais was held by the English at that time.