Yes, you should say something: he's being a douche, although maybe unintentionally.
Being a puppy owner myself, I understand how attached anybody can get to a dog. My puppy and I are pretty close, especially since I work from home, so we spend a lot of time together. But not everybody likes dogs or fully gets (or develops) that attachment. So for him this could be like your car died and you're getting a new one.
That said, though, I'm assuming you were pretty clearly upset when Jax died, and he didn't seem to notice. So explain how you feel about the whole situation, and how uncomfortable those jokes make you feel. That should clear things up. If not, teach the puppy to attack his crotch: that should do it.
HaHaHa! The last line's awesome. I should tell my best friend to do that with her doggy.
You absolutely should say something. Tell him he is being a rude, insensitive jerk. If he doesn't get it, the next thing you should say is "buh-bye!" For good.