Are there any accounts of the wise men story outside of scripture? The wise men were educated and would have documented their experience in detail. Also, they would have shared the Christmas story with everyone they came in contact with on their trip home. Once they were back in their home land their story should have spread far and wide.

Outside of scripture, are there any writings telling of the wise men story? If they indeed were educated, respected, and had influence and integrity, there would be non-biblical accounts. Have any been found?
Briefly, no. But I think you're making a number of assumptions here, which are not necessarily warranted.
We have very little information about the wise men. All we hear (from Matthew) is:

Magi from the east arrived in Jerusalem, saying, &quot;Where is the newborn king of the Jews? We saw his star at its rising and have come to do him homage.&quot;
They were overjoyed at seeing the star, and on entering the house they saw the child with Mary his mother. They prostrated themselves and did him homage. Then they opened their treasures and offered him gifts of gold, frankincense, and myrrh. And having been warned in a dream not to return to Herod, they departed for their country by another way.
(Matthew 2:1–2, 10–12)

We don't know where they came from, what they knew, how important they regarded Jesus to be in &quot;the grand scheme of things&quot;, or anything else.
You assume that the wise men would have recorded their experience in detail, and shared the story with everyone they met on the way back home. I don't feel so certain of this. Mary, for example, &quot;kept all these things, reflecting on them in her heart&quot; (Luke 2:19). Certainly the shepherds didn't, but we can't necessarily say whether the wise men were more like the shepherds or more like the Mother. Similarly, we shouldn't assume that they would have spread their story (they certainly wouldn't have wanted it to get back to Herod's Judea), or that others would have believed them or thought their story of any importance. We just don't have sufficient information about them or their positions to draw a conclusion like that.
All we can say for certain is that there don't appear to be any surviving eastern (e.g. Persian) accounts of the birth of a king of the Jews at that time.