Back handsprings come from repetition. I suggest you make stations working on specific pieces - the sit and jump, arms to ears, pac-man w/ spot, and hand spotting down a wedge or slanted mat. Remind them, push the floor down and away then reach for the ceiling (only then, if they want, can they look for the floor).
Front handsprings are a little tricker. They require a good amount of shoulder flexibility. Properly shape their bridge position, and hand spot "rocking" to standing from the bridge. 3 rocks or so per turn. Walking up the wall (like the back bend practice) will help as well.
There are a few ways to go from here. I had one suggestion that said practice a floor (or neck) kip to bridge, then stand, but I find this really tends to put a lot of strain on the shoulders and chest if the gymnast is not ready. What is good is that they can do this on their own, and if they need support they can do it down a wedge. The bad is potential for injury if they are not flexible enough, or do not bridge properly.
The other suggestion is pretty similar, but is a little more progressive. Start by doing headstands on the top of a wedge, and fall forward (ala headspring), to stand. Eventually, do a real headspring, and then when the hip action is understood, move to handsprings.
In truth, front handsprings have always caused me trouble. Any other suggestions would be wonderful! Though Im sure they will be similar to the ones above.