https://en.m.wikipedia.org/wiki/Castilian_War
I assume some of the Mesoamerican soldiers stayed in the Philippines and maybe took wives. Do we know what happened to them? Did they leave any cultural legacy that was not Spanish but rather Aztec, Mayan, or Incan? Did these soldiers wear Spanish style armor or was a Jaguar Warrior facing off against a Jannissary a possibility?
No chance of that face-off ever happening. Bear in mind that by 1578 the viceroyalty of New Spain was already heavily hispanicised, at least administratively: by 1555 there was a viceroy, a Royal Audience of Justice, a Royal Mint, a University, hospitals, barracks, etc. The soldiers would have worn the usual arms and armour one sees amongst the Spanish troops, even if part of the troops were of native origin.
As for the Incan and Mayan troops, I have my own reservations against that. There were indeed native american men amongst the Spanish garrisons and units in the viceroyalty of New Spain, but having Peruvians up there is just beyond the realm of plausibility. Mayans, I also doubt it. The Mayan area was not fully conquered until the late 17th century, and the accesibility was very complicated, that's why it took so long to pacify and secure the control of the zone. The possibility of the Captain General of Mexico recruiting troops in the Yucatán is a rather unfeasible idea. The native american troops within the Spanish army would have likely been from the area of the Valley of Anahuac, so mexicans, tlaxcalans, and possibly some tarascans, though they are not from that valley.
I invite our friend u/yawarpoma or anyone else to have a look at this
/u/TywinDeVillena and/u/Yawarpoma have covered the main points - the 1570s are much too early, and both Andean and Mayan troops are completely improbable.
There were transports of forced laborers and convicts to the Philippines, some of them indigenous, but in small numbers and mostly starting from what I've seen in the early 17th century. This included Nahua groups but other native groups would have been certainly possible. The much more important forced transport though was of Asian people via Manila to West Mexico. With slavery of Asians there lasting into the 18th century.
I've written before on those topics including Asian contacts with colonial Mexico - with a focus on the Aztecs - that you may find interesting, esp. the 1st one with some discussion of the Nahua author Chimalpahin:
more directly on Aztecs/Nahua in the Philippines
Hope this helps!