if archer can combine with breaking news so the runner doesnt lose the two tags and the end of turn.

can the same thing happen with joshua B, take the click trash to pawnshop and then not get the tag since joshua B isnt in play at end of turn?

No, it doesn't work the same, and it's because of a subtle difference in wording.

Breaking News says:

When you score Breaking News, give the Runner 2 tags.

When the turn on which you scored Breaking News ends, the Runner loses 2 tags.

There are two separate triggered effects on this card. One triggers when you score it, and the other triggers when your that turn ends.

Joshua B says:

When your turn begins, you may gain [Click]. If you do, take 1 tag when this turn ends.

Here, there is only one effect, and the consequences for triggering it are set up at the moment it triggers. (We've already seen and example of the phenomenon of a triggered effect setting up an effect to happen in the future - Chum.)