The funny thing is, I've never had anything bad happen to my game when I used to play around with my AR. It would freeze, yeah, but it doesn't wreck anything.
AR does not randomly cause Bad Eggs to appear. It would be the owner's fault for inputting the cheat wrong. Bad Eggs or ???? will appear in games if you set the Wild Pokémon Modifier to 0, an integer, or a number greater than 493. There's also an extremely slim chance of a Bad Egg appearing if you use the cheat to steal Trainer's Pokémon. Ironically enough, you can even find a cheat somewhere to FIX the Bad Egg.
And no, Action Replay cannot mess up your DS. Your DS simply reads code from the card it's given; it can't break from reading code, that just makes no sense.
Folk tales about these things are far and many, and 99% of "Action Replays are evil!" stories are fake, so do your research. One thing AR will do is break itself. The cheat card is very fragile, and to be honest, I wouldn't buy one only for it to break a couple days later, given the fairly steep price tag.