I don't see how this movie can be taken as any kind of "reboot", other than a desperate attempt by the fans to take a shot at X-Men: The Last Stand and X-Men Origins: Wolverine.
Except for the timeline regarding Xavier's use of his legs, the continuity of this film lines up with the other movies way too well to be thought of as any kind of reboot in any way.
The opening sequence is ripped directly from the opening sequence of the first movie. That's more than a "nod".
One of the focal characters of the movie (Mystique) looks exactly the same as she does in the original trilogy.
The one character who does look different (Beast) already has a built in explanation to why he looks different. The most obvious being Hank's explanation of how the "cure" worked on him - enhancing his mutant genes instead of attacking them. Given that his serum was taken from Mystique's genes, it's very easy to believe that his appearance could alter over a course of 40 years. Especially considering he is a mutant who is very insecure about his appearance, and given that he went from a scientist to a member of the President's cabinet, it's also believable that he would take measures on his own to alter his appearance.
Going back to Mystique, the explanation of how her mutation works is also a built in explanation as to why she still maintains a youthful look in the original trilogy, 40 years later, including in X-Men: The Last Stand after she's been cured.
Her relationship with Charles in X-Men: First Class can be seen to be a reason why she is so concerned when she learns of Stryker's planned attack on Xavier's mansion in X2. It could also help to explain why she'd turn on Magneto the way she did in X-Men: The Last Stand after she was cured and Magneto abandoned her. It doesn't feel so out of place now, knowing what she was like before Magneto got into her head. She was actually a good woman, with a good head on her shoulders and a good sense of morals.
Then there's the obvious - the way she reacts to Senator Kelly, as well as Nightcrawler, in the original movies, and the agents in X-Men: The Last Stand are reinforced.
The very fact that the timeline would prevent them from using any of the original X-Men (outside of Beast, who was obviously depicted as older than the rest in X-Men: The Last Stand), and as such, none of the original X-Men appear in the movie (if it was truly a reboot that disregarded the established continuity, there'd be no reason to exclude Cyclops, Iceman, or Jean Grey. And if you're using the argument that they are keeping Singer's films, and disregarding Ratner and Hood's films, then they would have been able to use Angel in the role, instead of Angel Salvatore).
William Stryker is referenced in the movie via his father being a part of the CIA, William mentioned directly by Xavier, and a reference to William's son via the woman with the blue and green eyes (perhaps William's future wife?)
The same Blackbird / X-Jet is used. The same Xavier Mansion is used. Rebecca Romijn and Hugh Jackman are both in the film, recognizing their original trilogy roles in this universe.
It connects to the original trilogy way too much to be considered a reboot. The only contradiction is the timeline regarding Xavier's ability to walk. That's IT. Everything else in the movie follows up with the rest of the series. And seeing as how X-Men Origins: Wolverine would have nothing whatsoever to do with X-Men: First Class, I don't see how they are throwing out the continuity of that movie either when nothing in First Class contradicted it (again, outside of the shaky timeline of Xavier's ability to walk).
There is no way this is a reboot. It's part of the same universe. Sometimes, a movie series has a movie or 2 that don't live up to the quality of the rest. That doesn't mean you scrap the whole thing. X-Men: The Last Stand and X-Men Origins: Wolverine sucked in your eyes. It didn't suck in everyone's. I know plenty of people who think those are the best, or like them just as much as the others. I know I do. But just because there are a couple slight continuity issues that were neglected because the film makers thought they needed to do so in order to tell the best story doesn't mean that it's a reboot.
Or is the Star Wars prequel trilogy a reboot too, because of certain continuity issues between the 2 trilogies?