I don't see any Boston Terrier in your pup.  Color reminds me of a Shih-tzu to me.  Could be another kind of terrier in the pup.  You could do a DNA test to get a general idea if you want.  They are not 100% accurate but may be helpful to you.

Looks like a Boston / jack Russell cross?

We can all take guesses & we all will see something different. I think it is crossed with another terrier cause of the terrier coat.

This is a pi$$ poor photograph showing very little of said dog - so we have very little to go on but it DOES NOT EVEN LOOK like a BOSTON!!!!!!!!  And black coat color genes DOMINANT over any other color so the dog NOT being black anywhere in the photo is hard to explain as having a BLACK (the major coat color Bostons come in) parent.  

Sure, if they knew the mother was one and they whelped it or had both turned in, then that could be right (more likely she was a Boston MIX herself) but it looks like she got bred by a what was referred to as a Jack Russell Terrier.  

Some of the JRT breed meeting certain height requirements have been renamed as a Parson Russell Terrier.  JRT can have 3 types of coat; smooth rough or broken.

https://www.pets4homes.co.uk/pet-advice/jack-russe...

Differences in the JRT and PRT:

https://www.canidae.com/blog/2013/07/difference-be...

https://www.luvmydogs.com/2017/07/what-is-differen...

There's not a way to know with any accuracy from his appearance. I've seen a similar looking dog that was half Boston Terrier and half Shih Tzu, but the mix could be with any longer coated small breed (or even multiple other breeds). Just enjoy him for the unique individual he is!

What KIND of dog?  It's a mutt - nothing more.  It's not a full bred anything and there's nothing in between mutt and pedigreed dog. But what does it look like?  Based on THAT lame photo?  We can't see his face, what his body is shaped like, how tall he is, how long he is. He's a MUTT - get over it.

You can only go with what the Shelter has told you  - Boston Terrier mix.  Does it REALLY matter .... and it's impossible to even see thatl he's even part Boston Terrier from that photo!  

With mixes, unless the breeder knows what happened, you really never know what the mix is, in any case, especially these days when more and more BYBs are messing around with long-established breeds.

It's a mixed breed. You know how I know? The paperwork SAYS SO !

Look, it looks like a cute dog, but, they told you, it's a mixed breed... love it for that...

That is not any breed in particular. That is a mutt.