Can a runner  have negative damage or does his health just stop at zero?

Example:

Runner is holding four cards.  Runs on remote server on his second action.  Encounters Junesong Provsion with three advancment tokens dealing 6 net damage.  Runner would have -2 cards in his hand and his two remaining actions would only get him to 0 cards causing him to flatline.

Is this how that would be played or is the runner simply just reduced to 0 instead of -2?

In your example the runner has lost the game. If the runner is ever out of cards and still has damage to take they are flatlined and the corp has won the game.

Ok I wasn't sure about this because the rulebook does state at the end of the runners turn.  This has never come up in one of my games but I was going to play that if the runner were at say at -1 with two actions left, he would be able to use the last two actions  to come back to 1 card preventing a flatline.

So what I am hearing is if the runner is ever negative even on his turns with actions to spare to get back to a positive number of cards, the runner loses.

Doogan said:

Not only.

Form rules, pag 20:

If the Runner takes more damage than the number of cards in
his grip, or if he has a maximum hand size of less than zero at
the end of his turn, then he is flatlined and the Corporation wins the game.

From rules, pag 30:

Flatline: A condition that results
from the Runner being forced to trash
more cards than he has in his grip, or
from having a maximum hand size that
is below zero at the end of his turn, and
which causes the Runner to immediately
lose the game.

Doogan said:

The "end of the runners term" quote you are referring to is in reference to a hand size reduction caused by suffering brain damage. 

Basically,if the runner ever has to discard a card from his hand and is unable to do so (discards down to -1 or lower), he immediately loses the game. 

If the runner ever has his hand sized reduced to less than 0 for whatever reason (currently only possible via brain damage), then he will lose at the end of his turn.