Yes, Pokemon GO players can catch the shiny version of Mesprit. However, like all shiny Pokemon, it is extremely rare.
There is a 5% (or 1 in 20) chance of any Mesprit you encounter in Pokemon GO being the shiny version. This is the standard shiny rate for Legendaries in Pokemon GO.
The best time to catch this ultra-rare shiny is during one of the two Pokemon GO Mesprit Raid Hours. This is because the number of Mesprit Raids will be greatly increased during these events,
However, it is important to bear in mind that Mesprit is still region-exclusive even during the Raid Hours. You can only encounter Mesprit in Europe, the Middle East, Africa, and India.

What Color is Shiny Mesprit in Pokemon GO?
The shiny version of Mesprit is bright yellow and red in color. This makes it appear very different to the gray-blue and pink of the standard version.
Even if both versions of this Pokemon appear very different, they both have yellow eyes and red diamonds on their body and forehead.
The shiny versions of the other two members of the Lake Guardians – Azelf and Uxie – are also available to catch in Pokemon GO. Although, they are also exclusive to certain regions:

If you are a shiny hunter, you might want to find out which other Legendary Pokemon have shiny versions. Check out our list of every Legendary Pokemon and Ultra Beast in Pokemon GO to see which ones are missing from your Pokedex!