Apparently it has been rumored there are more secret characters than just Yuffie and Vincent. Below is a list of them.
SEPHIROTH
Supposedly you can get a clone of Sephiroth to join you.
ZACK
It has been rumored you can also get a clone of Zack, Aeris' ex-boyfriend, to join you. I don't know the process, though.
TEOIH
Yep, another secret character has been rumored. This time you can get Teoih, the mysterious guy who rides the black chocobo at the Chocobo Races. Go to Wonder Square at the Golden Saucer and play the very ordinary fighting game that's impossible to win. Now do the impossible. Win the game and Teoih will appear, congratulating you. He'll then ask if he can join your party. Not only that, you can also get to ride his black chocobo when you're on the Overworld. It's not supposed to do anything special, but it has good stats when you race it at the Chocobo Races.
BOXER
This rumor has interesting origins. Apparently Boxer is supposed to be a goblin you can have to join your party. Go to Goblin island and eventually you'll fight a friendly goblin. Use Goblin Punch on him and he'll join your party.
CHOCOBO
Yes, it has also been rumored you can also get that lovable bird to join your party as a fully-fledged party member. Again, I don't know the process.