The original Japanese one is in fact a hack/generated Pokemon. HOWEVER! There was no way for you to have known, prior to uploading it on Pokecheck. The only thing out of place are the trash byte terminators on the OT, and nothing aside from Pokecheck actually checks those. So, there literally was no way for you to know, because the game would accept it on Wifi, and everything else.
The Eevee that have been breed from it are 100%, non-hacked Pokemon. When a Pokemon is bred, even with a hacked Pokemon, the game always generates a valid Pokemon. You did nothing wrong, ignore the twits that make comments in the chat. In the future, you are going to know to check the Pokemon you download. No harm, no foul.
Your other question was how to fix it. If you look at the trash bytes for the OT on the Japanese Eevee, you will see it terminates with FFFF. This is incorrect for a DW Pokemon. Every DW Pokemon I have ever caught has always had the OT terminator as 0000. I uploaded the Japanese Eevee to Pokegen, and changed the terminator: http://www.pokecheck.org/?p=detail&uid=348382. As you can see, it no longer will show as hacked. (Also, I'm not condoning the use of this Pokemon, I'm simply trying to show you what was wrong, and why it was hacked.) Trashbytes are a very complex thing, and even I still know little about them.