https://c5online.pica.gov.jm/apply/
I used Firefox for a browser and still got the privacy warning. Just click "advanced" and then go to the bottom of the page where is says "accept the risk". That worked for me.
When you fill out the form on the plane, the immigration officer has to input all the details into his computer and that takes time. When you do it online, all the information is loaded already and appears when he/she scans your passport. Maybe a few glitches still but when everyone goes online it should seed things up. Also, first rule when landing at MBJ, never ask airport staff where to go or what to do. They haven't even got a clue what day it is. Chances of getting a right answer from them is 0%