The name Yeshua is an Aramaic form of the name Yahushua. The problem with calling him Yeshua is that it is not his name!
If someone has the name “James,” that is their name. The Spanish form of this name is “Diego,” or “Iago.” Their name does not change to “Diego” once they cross into Costa Rica or Colombia. Yahushua’s name was prophesied long before he was born.
“Take the silver and gold and make a crown, and set it on the head of the high priest, Yahushua son of Yahuzadak. Tell him this is what YaHuWah of the armies says: ‘Here is the man whose name is the Branch, and he will branch out from his place and build the temple of the Lord. It is he who will build the temple of YaHuWaH, and he will be clothed with majesty and will sit and rule on his throne. And he will be a priest on his throne. And there will be harmony between the two.’ (Zechariah 6:11-13)
The King-Priest who is the Branch (a title for the Messiah) would have the name Yahushua (יהושוע read right to left is YH-oo-Sh-oo-uh), just like the son of the High Priest. That is a Hebrew name, not an Aramaic name.
When we come across the name of the Messiah in the Greek scriptures, it is spelled Iesous. This is also the name given to the son of Nun, the 2nd in command to Moses. They share the same name! We know that when we come across Iesous in the Greek it is really the name Yahushua for both the son of Nun (who people call Joshua) and the son of YaHuWaH.
Yeshua is not a Hebrew name. He was not prophesied to have the name Yeshua. Other people with the same name were not called Yeshua. His name is Yahushua! YaH-oo-Sh-oo-uh! Yahuwshuwa! YaHuWaH is deliverance!
Many people believe that people in Yahudah (Judea) spoke mainly Aramaic, and so believe that Yahushua and most of the people spoke Aramaic and not Hebrew. This theory has largely been dis-proven. Archaeological evidence in Yahudah points to Hebrew being the main language of the people. Writings from the area dated to the 1st century skew heavily towards Hebrew. Josephus, a 2nd century historian, commented that most of the Hebrew people spoke Hebrew and that their religious leaders forbid them to learn other languages.