You can do either way, but down is always much easier. But instead of removing the boxes simply remove all the doors and hardware will be the way to do it if they are currently up.
Also stain is really left to unfinished wood surfaces. Paint is a faster way to go.
What color or look are you trying to achieve?
Many folks use a gel stain over existing stain to darken the wood. Here's a few Hometalk members that have done so....
If your cabinets have a sealer or varnish on them, the stain will just sit on top. Stain is meant to be absorbed into raw wood.
@Kass, is correct, but there are products out there that do work over prefinished wood. Personally I don't think they look as well as it does when one removes the old finish. I was just giving an idea of what others have done. Minwax actually has directions now for using their Polyshades over previously finished wood. http://www.minwax.com/wood-products/one-step-stain-and-finishes/minwax-polyshades
Though they (minwax) recommend to removed the old finish before using their gel stains I know many have used them over previously finished wood with results they were happy with. Like I said, I prefer to start with raw wood. I highly recommend whatever the original poster chooses to do they do a test first somewhere inconspicuous. Like on the inside of a seldom used door.
Thank you ladies so much. I do have a very light wood in my kitchen. Mrs. Becky and would like to darken it up. Just not my taste, However it does have a varnish on them. I was told that I could use a mineral spirit to strip the stain.. Is that correct? Kass thanks for the info. I dont want to make a mess of things..
If it has a varnish on them and I wanted to strip before I restain. Can I use mineral spirt to strip if I have real wood? My cabintes are light and I want to go darker.
Mona, without seeing photos of your cabinets I can't give you a definite answer. I'd suggest you remove a door and take it with you to make sure you buy the correct product for removing the old finish. If the doors are flat you and are only varnished/clear coated/no stain you can just sand instead of using a chemical stain remover though.
Becky has this right. Take the door to one of the specialty paint stores (like Ben Moore or Sher Wms). They are trained to know which products to use on which surfaces.