In short, no. Although it was advertised as such in order to facilitate "name recognition" and sell games, Chrono Cross is, in fact, not a true, direct sequel to Chrono Trigger. It is actually a very loosely-adapted remake of an earlier Japanese game called Radical Dreamers.
As Chrono Cross scenario writer Masato Kato said in an interview with Weekly Famitsu:
Q: "First of all, this game doesn't seem to have the name 'Dream Project' associated with it. Is it truly a sequel to Chrono Trigger? And why isn't it called 'Chrono Trigger 2'?"
A: "There's a long story behind that, but to keep it short... After finishing Chrono Trigger, we created a game called 'Radical Dreamers'. Although we hadn't planned it that way initially, Radical Dreamers eventually ended up as a sort of sequel to one of the subplots in Chrono Trigger which we weren't able to tie up in Chrono Trigger itself. When Radical Dreamers was finished, we did Xenogears, and when talk turned to what to do after that, we decided to redo Radical Dreamers properly. That's why we didn't give the new project the name 'Chrono Trigger 2'."
There are also these interview quotes from Chrono Cross producer Hiromichi Tanaka (originally featured on GamePro.com):
Q: "With Chrono Trigger (for SNES) considered a classic by many RPG players, was there a concern that not many references to the original are present in Chrono Cross? Were you concerned that you might alienate loyal fans of the original?"
A: "As I mentioned before, Chrono Cross is not a sequel to Chrono Trigger, so I'm not worried...On a different note, the original scenario writer for Chrono Trigger, Masato Kato, worked on Chrono Cross's scenario as well, but actually, there's another game he worked on, called 'Radical Dreamers,' which was released between Chrono Trigger and Chrono Cross. This was an adventure game released on the Super Famicom online gaming system called the Satellaview. Radical Dreamers served as the bridge between Chrono Trigger and Chrono Cross. Gamers who have played all three games can probably figure out the connection, but since the media itself was so unique, I don't think many players know about it. As a result, we had to make sure players could play Chrono Cross without being too conscious of its connection to Chrono Trigger. This is why we have the title 'Chrono Cross' instead of 'Chrono Trigger 2.'"
Q: "The original Chrono Trigger seemed to leave little room for a sequel. Was the success of the original title unexpected?"
A: "I seem to be repeating myself, but Chrono Cross is simply a new title..."
Thus, as the Chrono Cross designers have repeatedly stated, Chrono Cross is in fact a remake/sequel to Radical Dreamers and not Chrono Trigger. The Chrono Trigger reference were most likely inserted as a cheap ploy to sell games. Anyone who disagrees obviously contradicts the Chrono Cross developers themselves and thus can be safely branded a fool.
Because of this, Chrono Cross is not the true official word on the fate of the Chrono Trigger characters. It is merely a non-canon, completely hypothetical "what if"-type scenario. Nothing more.
