I have been a few times. One time my friend picked me up at Jamaica Train station. Another time I paid some guy to drive me from the train station (I wouldn't even call it a gypsy cab, it was just some guy in a Toyota...I had to move his kid's books out of the way in the back seat).
What is the closest train station (subway or LIRR)?
You can also take the E or F to the Union Turnpike stop but from there you would still need to take the Q46 bus to the campus. The SJU campus is not directly accessible by subway
If I want to get a taxi/livery cab from the train station, you think the best bet is Jamaica Station?
Yes...walk down to street level (takes a bit) from the Jamaica platform and there will be cabs all over from people being dropped off and taken. Why there is no subway stop right in front of SJU is beyond me.