When you introduce two kittens to another cat, should they start to use the same litter box?

Answers:
Sure, I don't see why not. If you're finding messes though, and the kittens are litter trained, it might be a good idea to get them another "poop deck". In most cases though, cats will all use the same one without any problems. The smell of the new cats shouldn't bother the old one.

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You should not make them use the same litter box. After they get used to eachother, then start doing that. Because if the cat you already had shares its litter box, it will feel like it is being replaced. That is what happened to my friends cat so yeaa...

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no

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every cat i've added to my household (went from 2 to 3 to 4) had no trouble sharing the litterbox.
my only suggestion is that you buy another one anyway, because 3 cats using the same box can get "crowded"

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No, the cat should have its own litter box...Some cats will never use the same litter box!

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If you haven't had the 2 new kids tested for feline leukemia or other sicknesses I suggest you do that asap, if you've already put them together with out testing then every one should have that done. Some like leukemia or feline version of HIV have no known cure and are highly passed on. If you got these lil ones from a shelter, don't assume they check for it, ask! Not all shelters do these tests, even though they really should. Assuming they're clean on health, I would start with how their personalities are, easy going and friendly cats/kittens are going to need less time on neutral ground then cats/kittens that are jumpy skittish of others. Keep in mind though some cats just like to be the only cat, they may tolerate them but they may never like them. Kittens usually get along well reguardless of who it is, they're very curious, playful, and harder to have fights over territory then older cats (even ones that are fixed will fight over their possessions and space at times). Supervised visits are best at first, that way you know whats going on and no one can be seriously hurt or at least in the event of a fight not as badly. Over time longer periods and alone time come, litter box use will usually play itself out. Just make sure you have enough to take care of every ones needs, more then one litter box is usually a very good idea since it helps with the space and in preventing kitten accidents in my experience. Clean litter is always a must of course. Any kind of food though is some thing that shouldn't be given while they're together, cats and kittens both become nervous/possessive and most likely hateful if not harmful to each other in trying to protect their own food.

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you should have one box per cat. they may end up sharing, but that's the rule. what if they need to go at the same time?

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I have always used seperate litter boxes when introducing a kitten into the house, at the start i noticed the older cat wouldnt use it if he smelled something that wasn't his.
Its fine after a while when they become friends.

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The kittens definitely should be separated from the cat until they get to the vet. Doing so will protect your current pet from various potential communicable diseases (viruses, worms, fungus, etc.)
Once that's done, you should consider keeping them separated for at least two weeks. Let them sniff each other through the door and on you so that they have a better chance for a smooth transition. Go slow. First impressions count.
That being said, there will already be at least two litter boxes in your home. You should really consider a third. That way if the older cat starts to guard the box (cats are territorial by nature and some just like to be bossy about that kind of thing), there are other litter boxes available, as opposed to your furniture or floor.