Hi Elaine, is it the island's countertop? Any kind of really serious glue would probably work ok, though if grout is involved you will need to replace the grout. Gorilla glue, liquid nails, masonry glue etc.
You need to scrape out the old adhesive apply, spread tile adhesive on the back of the tile, put it in then use spacers in the corners to keep it in place. The final step is to use grout the tiles. I found a video that shows what to do, but the first part is how to take out a cracked tile, which you don't need. The process for putting in the tile starts at about 1:27. I hope this helps you. Wishing you the best.
You can use tile adhesive, scrape off as much of the old as you can, reapply to the back of the tile and press firmly. Then you'll probably have to regrout as well.
If you are not really "in love" with the tile, consider maybe put a butcher block on the island.
I made an island from a dresser and put butcher block. It is so much easier to maintain and actually more sanitary than tiles and grout on a counter.
I chose to put a few coats of water based poly on mine, because it is not used as a cutting board. I do not enjoy maintaining the wood with oils or waxes.
I got my slab from Home Depot, 30 x 72 and 1 1/2 thick (all in inches) for a work bench. I paid somewhere around $160 for it.