People having problems housebreaking their beagle can get frustrated and desperate and they will wind up trying some very strange ideas that seem good at the time but have long term negative effects. litter box training is an example of an idea that seems like a good temporary solution but unless you are prepared to possibly make it a permanent solution then do not use it.
The toy breeds of beagle, usually cross breeds with smaller dog breeds, can probably get away with a litter box but you are going to need to be realistic before you employ the litter box solution. Your regular sized beagle is not a good candidate for a litter box. If you are having housebreaking issues with your regular sized dog then try something else other than a litter box as your full sized dog will choose what option makes them comfortable if you give them an option and you probably will not like their selection.
Pocket beagles and dogs that are a cross with smaller breeds like a Chihuahua are probably your best bet for litter box training and even with the smaller dogs it is not going to be easy. Do not kid yourself into thinking that it will be an option for your beagle, once you introduce the litter box the dog will probably prefer the litter box in the warm house over going outside in the cold. So once you get them into a litter box you will probably not get them out.
A dog is not a cat and while that may seem obvious it needs to be said. Just because a cat can be taught to use a little box does not mean any animal can learn it. Cats that come from indoor cats are taught to use a litter box by their mother and cats have an instinct towards the litter box. Dogs lack this instinct so you are going to have to train them and there are two situations that would require use of a litter box and training in each situation is different.
If you are bringing home a pup for the first time and you make the decision right up front to use beagle litter box training as the way your dog will be relieving itself then you will need to teach it as a pup. Get your beagle used to the box by putting it in the box and then using positive reinforcement like petting and positive talk. If you notice your puppy giving the signs that it is going to relieve itself then put the puppy in the box and if it uses the box then give it positive reinforcement again. If you do it with the right touch it should only take a week or so to train them.
If you are using the litter box as a last resort for a smaller breed that cannot get the hang of outdoors then you are going to need to place the litter box in their favorite spot for relieving themselves in the house and then spend about two straight days with your dog constantly putting them in the litter box and using positive reinforcement. If it works then you have a litter box dog.