Can a grounded fig that doesn't have leap/climb or phasing move thru an exact diagonal that has hindering terrain on one side and an enemy fig on the other side and keep moving or would it have to stop?

It would have to stop:

The fig would become adjacent to the enemy fig. Since you have already stated that the fig does NOT have any auto-breakaway powers (assuming they don't have HSS...), they would have to stop immediately after becoming adjacent. Remember that adjacency covers ALL EIGHT squares around the enemy fig, including diagonals.

Also remember that if he were to travel through two squares of hindering terrain via a diagonal, the two squares are considered continuous hindering terrain, and while he could cross through, he would have to stop IMMEDIATELY after doing so, and his move would be over.

If you are presuming that it is a HSS character that is trying to move between an enemy character and hindering terrain and a diagonal, then that would also be considered hindering terrain, because the enemy fig is effectively blocking terrain.