Here's the In Game Proof That Pokemon Stars Is On the Way
Missing Pokemon Gyms
One of the biggest changes to Pokemon Sun and Moon was the lack of Pokemon gyms. The game replaced gyms with Island Trials, a test that mixed Pokemon battles with a few simple puzzles.
However, gyms could make a big comeback in Pokemon Stars. Almost all the cities in the Alola region had an empty space or two, all of which were the perfect size for a new building. The popular speculation is that those slots will become Pokemon gyms in Pokemon Stars, as there seems to be too many empty spaces to just be a coincidence.
Necrozma
After beating the newly formed Pokemon League, the player character is immediately called into action to capture multiple Ultra Beasts that have popped up throughout the Alola Region. After capturing/defeating multiple Ultra Beasts, the player heads to Ten Carat Hill to encounter a strange and powerful Pokemon named Necrozma.
There's a lot we still don't know about Necrozma - its PokeDex entry hints that it's an ancient Ultra Beast that has slept underground for generations, but that's never really established in game.
Sequel games usually flesh out the "third" Legendary Pokemon in a set, so we're guessing we'll learn more about Necrozma's history and abilities in Pokemon Stars. However, we'll note that Zygarde (the third Legendary Pokemon in Pokemon X & Y) didn't get much of an explanation until Pokemon Sun and Moon, so it's not a guarantee.
A Third Lycanroc Forme
Lycanroc is the evolved version of Rockruff and has two formes: a Midday Forme that can only be obtained in Pokemon Sun and a Midnight Forme that can only be found in Pokemon Moon. However, there's evidence that there could be a third forme also hidden in the code.
A dataminer discovered that there's a placeholder space in Pokemon's Global Link service for a third Lycanroc forme, which presumably would be exclusive to Pokemon Stars. This third forme isn't technically part of Pokemon Sun and Moon, but since it was found in Pokemon Global Link, we'll include.
The same dataminer also found evidence of alternate formes for Mimikyu and Fearow as well, so it's possible that these could be the "new Pokemon" mentioned in the Eurogamer rumor. We find it pretty unlikely that a Pokemon Stars game would introduce actual new Pokemon, but a few new Pokemon formes definitely wouldn't surprise us.
The Wierdo Pokemon Bank Cutscene
Game Freak added a strange Easter egg for fans who used the Pokemon Bank app to transfer their older Pokemon to Pokemon Sun and Moon. There's a strange man in the Haina Desert who requests to see either a Solrock or a Lunatone, two Pokemon that aren't native to the Alola Region.
If you transfer a Solrock or Lunatone over to Pokemon Sun and Moon and show it to the man, he'll reveal that he's actually an extraterrestrial and bestowed "star pulses" to several men in the distant past. That's a reference to a group of characters in Pokemon Omega Ruby and Alpha Sapphire who granted the player special O-Powers.
This likely is just a bizarre Easter egg, but some fans thought the references to "star pulses" was also a nod that a Pokemon Stars game is on the way.
Decreased Second Screen Functions
The Pokemon franchise had made good use of the Nintendo 2DS and 3DS's second screen system for years, but Pokemon Sun and Moon's second screen functions noticeably took a step back. In Pokemon Sun and Moon, the bottom screen was typically home to the RotomDex, which usually served as an in-game map telling players where to go.
Most of the menus and secondary features (like Poke Pelago or the Festival Plaza) required either button use to access or needed the top screen to function. While many of those features required a touch screen, none of them required the use of both screens at the same time. The second screen only really came into play during Pokemon battles, although some fans thought that could easily be switched over to an overlay of some kind.
So how are the second screen functions relevant to Pokemon Stars? Well, if the game is coming out for the Nintendo Switch, it'll only have one screen with which to operate. There's a belief that Pokemon Sun and Moon was Game Freak's way of slowly phasing out the second screen in preparation for a complete transfer to the Nintendo Switch.
The Caveats
Looking through our list, there's a lot of stuff that could either be coincidence or teases at a future game. A sequel game isn't a sure thing, as there were plenty of hints that Pokemon X & Y was going to get a sequel that never panned out.
Nintendo has a history of announcing a Pokemon game during the second week of May, so if we are getting a new Pokemon game in 2017, we'll almost definitely find out this week. If we don't get news now, don't expect an E3 announcement, as Game Freak has always preferred the Pokemon franchise to get its own spotlight.