There is no typing line for this question. A Beagle Boston Terrier mix is called a Boglen, who is now officially a hybrid or "designer dog" in the registry of the American Canine Hybrid Club (ACHC). Each hybrid dog must have purebred parents. When you get a Beagle-Boston Terrier mix, you get the characteristics of both purebred parents.
You will need to ask a breeder for photos of his Boglens to know the characteristics of his Beagle-Boston Terrier mix. The snout of Boglen tends to be longer than the short muzzle of Boston. The Boglen's ears tend to be sharper up in that cheerful look of Boston. The coloration of the Beagle-Boston Terrier mix may resemble that of Boston, so the dogs are either black and white, brown and white, or brindle and white. Or you could get a Beagle tricolor look with black, tan and white. Whatever the dominant features, the Boglens are excellent dogs.
If you search Boglen on the web, you will not find the direct description of personality characteristics like that of standard dogs. The personality of a hybrid dog can be any assortment of traits of his parents. Some people who have a negative view of the world will say that you can get the "worst" personality traits of each dog. Any dog lover will say that it's just ridiculous because the dog's personality can not be described in terms of "worse". Some dog lovers love some personality traits more than others. The Beagle-Boston Terrier mix will have the characteristics of both parents. If you love Beagles and Boston Terriers, you'll love a Boglen.
Hybrid dogs are not "dogs" because their lines are clear. With each parent, a purebred dog, you will certainly get the characteristics of these two breeds. Many people have very positive things to say about their mixed breed dogs. Some people say that the "all American" has fewer health problems. You definitely do not take any risk when you buy a Boglen.
Some people say that boglens are easier to train than beagles. This might be true, because all those who have a Beagle will tell you that they are certainly not easy to train. The Boston Terriers are renowned for being very smart and training. If you want to have a Beagle-type dog that does not bark as much, you may want a Boglen because the Boson Terrrier is not really a barker. If you love happy dogs that like to play, you will definitely be happy with a Beagle-Boston Terrier mix.