Can I still play Unikong?

Yes, you can still play Unikong through this link here. Not sure who uploaded it, but I have played it a few times there and everything works fine.

This is a really soft answer, but if the other April Fools are any indication, you may still be able to play the game after Unikong is open sourced. Stackegg was released on Stack Overflows Github. I would expect Unikong to be released in the same fashion.

Take a look at this question on the Stack Overflow Meta. From this answer someone uploaded a bit of the source code to their own site so that you can play the game there.