the eyes drop from the beasts that look like big bats, any area in that island that has bushes/beasts that cost 8 energy has the bats. eyes drop perhaps from every other bat.
you perhaps have to complete the sweltering jungle squares puzzles to open/unlock other parts of the island. i looked at the sandy shore and that will have eyes

the quest is a repeatable quest that is essentially area wide. it has nothing to do with exiting from that square. to exit that square and pass on to other parts of that island you simply do what you do everywhere else, complete the puzzle and possibly get the custom items too (i can't remember as it has been 4 years since i have played bw1)