A little salt, a little pepper, and a big frying pan.
But seriously.
I did this as well when I played through it; my character couldn't help but feel bad at the wanton murder of unborn creatures. So, she rescued one... and then there was some DM shifting at my FLGS and my character got skipped ahead to another point in the game. Oh well.
There are a few options with dragon eggs. First is trying to tame them; it's easier to do it when they're young, but this is still a black dragon we're talking about. It's not going to be easy, assuming we're even entertaining the notion that it's possible. Starting from an egg is just the best-case scenario. Also, I'm pretty sure the Tyranny of Dragons campaign doesn't go terribly long, so it's not like the hatchling is going to mature very much during the story, which means the party won't get much/any use out of a mature dragon. Hell, they'd be lucky to get a Wyrmling by the end of it. And that's assume they manage to keep it and train it.
Another option, and possibly the easier one for the DM, is to find a collector in a city who would pay top coin for a dragon egg. Price them at about 500 a pop, which is a lot of coin at that level. Greenest probably doesn't have much to spare, but I recall some other large cities being on the narrative horizon.
I seem to recall there may be some other information in the module, regarding what the eggs need in order to gestate and hatch. By taking them out of their environment, they will remain dormant. If that's not in the module (and my DM was just making it up, as a DM is wont to do), then use it anyway. Again, easy way to avoid having to deal with a quasi-friendly black dragon owned by the party.
Of course, if you really want to go crazy with this idea, then allow the players to keep the egg, hatch it, train it, and so on. But, behind the screen, remember that they are black dragons and they are loyal to Tiamat. Who may or may not be rising in the near future. Sure would suck if all that hard work turned against them at the last moment, wouldn't it?