Okay... I don't consider this exotic (unlike pumpkin pies), but you probably do so here's how to prepare candied squash/pumpkin.

Ingredients:

• 1kg of squash. Any sweeter variety works*. Only the flesh; no seeds or skin. Cut it into large-ish cubes (~4 cm should be good)
• enough water to cover the above.
• 10g of food grade quicklime, or roughly a tablespoon.**
• 500g of sugar.
• 500ml of water. (yup, again)
• whole cloves and cinnamon sticks to taste. 3~4 cloves and 2~3 cinnamon sticks should be enough.

1. Put the squash cubes into a bowl, cover them with water, and add the quicklime. Mix it a bit, and let it rest for 3h.
2. Drain the water and rinse the cubes. Then use a fork to pierce a side of each cube (so the syrup penetrates it.). Reserve.
3. Boil the 500ml of water. Add sugar, cinnamon, cloves. Let the sugar dissolve.
4. Add the squash cubes, and cook them in the syrup, over low fire. It needs to be low fire, otherwise you won't be able to cook them evenly.
5. Keep cooking them for 1h or so, mixing it occasionally. Be gentle, as you don't want to break the cubes. The syrup should reduce quite a bit, and the cubes should be soft on the inside; some leathery skin is expected (and desirable), but if they're still tough and the syrup reduced too much it's fine to add a tiny bit of water to compensate the loss.
6. Let it cool and enjoy. They should turn out like this:
   

Notes:

* traditionally this sweet is made with this sort of squash, known in Portuguese as "abóbora menina":

You can use pretty much any sweeter variety of squash though. Kabocha, pumpkin, buttercup etc.

** using quicklime on food might sound weird, but it's fairly common across the world. For example they use it in North America to nixtamalise maize, and in China for century eggs. I don't think that you'll have a hard time finding it in Korea.