Yoshi as a unique strength in this matchup, given that DJ has armor, we can jump out of combos a lot earlier than most, or even nair out (much like you may see in a Luigi ditto).
Personally in this matchup, I find the best way to edge guard is trying to snipe with an egg before trying anything else, otherwise you could take a green missle, or if you're unlucky, a misfire to the noggin. Luigi's down B is something to look out for too, since if you chase him into the air he can hit you with it and if you prematurely try to punish his landing he can hit you with it. But this can also be a good thing, since against a lot of Luigi players, it isn't hard to bait this move out by simply faking a landing punish with a run up shield or just dashing through. If he learns this early, you might scare him from using it for a while, or if you want to keep him conditioned so you have an easy kill setup later, that works too.
I'm not as experienced in the matchup as some of you might be, but the major points in this matchup to me seem to be; number one, don't get grabbed, number two, if you do get grabbed, stay calm and either DI, jump or nair out of the down throw combo. Number three, bait tornados for the punish and punish any movement screw ups where Luigi skids, even if it is only with an egg. Number four, be careful about gimping and use egg sniping to your advantage.
If I'm wrong about something, somebody please correct me, I don't want to be giving out false information.