The two questions you should ask are:
whether it's the player or the character that's uncomfortable with your proposed shift in mood, and
if it's the latter, whether your players would be OK with the kind of intra-party conflict the shift would generate.
Is the player really OK with it?
If your player is uncomfortable with the campaign turning evil (or at least morally ambiguous) and just "doesn't want to go there", then I'd generally suggest not going there. You started the game with characters that are noble heroes, and all your players seem at least OK with that — you shouldn't change that in the middle of the campaign, unless you're sure your players will be OK with the new theme, too.
If you do still want to try the mood shift, please do make sure to talk things out clearly with the objecting player. Even if they say that it's their character who wouldn't go along with an evil plot, they could be using that as a cover for personal unwillingness to explore such themes. Just make it clear — preferably in private — that they can say "no" if they want, and that you'll respect it.
(Also, no discussion of game themes and intra-group tensions would be complete without a link to the Same Page Tool, so here it is.)
So the player is OK with it, but their character isn't. Now what?
So you have five nominally good characters who are increasingly tempted to "cut corners" and use questionable means to achieve their ends, and one who refuses to do anything of the sort. That's a great recipe for lots of tension between the party members.
This can be a good thing or a bad thing. It can be bad because it requires a tricky balancing act between party cohesion and inter-character conflict, and maintaining that balance can require some skill from both the GM and the players.
On the other hand, it can be good because, done well, it can be really fun to play and makes for great plots.
Note that such conflict doesn't necessarily have to turn into an actual PvP fight (which, based on what I've heard, would generally be a bad idea in 4e anyway). In fact, I'd suggest that, at least 99% of the time, this is something you shouldn't let happen, just like you generally wouldn't let the party get all killed by falling rocks or let the characters just split up and go their separate ways. It just doesn't make for a fun game.
Instead, if you want to try this, you and your players should agree in advance, out of game, to never let things get quite so far that a fight (or a party split) would be unavoidable. This means that the other players — and their characters — need to stay aware of the fact that they need the "pure" character's help (for some reason; preferably a plausible one) and that they therefore have to stop just short of doing anything that this character would find totally unacceptable.
It also means that the player playing that character has to be OK with the fact that their character also, for some reason, needs to cooperate with the others, even if he (the character, that is!) is feeling distinctly uncomfortable about it.
This does require a certain amount of "meta-game" communication between players, and possibly some creative adjustment of character behavior or even plot events to avoid escalating the tension too far. Do tell your players that it's perfectly OK to, say, ask another player out-of-game how their character would react to a particular action by another character, or even to just raise a hand and say "Guys, please don't do that, there's no way my character wouldn't righteously smite you if you did it."
Tell Them Your Goals
If you haven't already, I would start by telling them essentially what you just said here. That there is no "one true plot". Tell them that introducing an evil person / problem does not make it the overriding campaign unless they want it to be. Tell them that you are willing to follow along with their character's background goals.
If they want to play that style of game, there is a good chance that will fix a lot (but not all, habits are hard to break) of it right there.
But its worth remembering that not everyone wants a sandbox. I prefer to play in games with good plots with well laid out characters. While there are some exceptions, GMs frequently are willing to invest more time into plotting and developping good NPCs if they are assured the PCs will focus on them. For that reason, I am perfectly happy to with what you might call "light rails". I want some character autonomy, but I am perfectly happy to follow "the one true plot" if it gets a better story and better NPCs.
By telling them your goals directly, they will either agree to go along, or directly ask you to put some rails back and then you can choose as a group how much of the game should be on rails.
Drop many plot hooks, make them choose
In the game itself, one way to force them to avoid rails is to drop so many potential plot hooks they have to choose which ones to focus on. You can even show that the world is a living breathing place by showing them the resolutions to the plot hooks they didn't take.
For instance, they sit in a bar and overhear of some great evil. They may be tempted to go after it immediately, and there are no rails, so they can! But then someone comes up and asks them for help urgently. And I don't mean Skyrim style urgently where everything is urgent but will wait for you without ever changing, give them an actual deadline measured in something like game days.
Now, they have a choice. Go after the big bad or help this person? In a game on rails, the answer is first help then go after the big bad. Obviously, the person asking for help is just there so they can level up. But you can break that by making it clear the big bad isn't standing still. If they go after him now, he won't have completed his McGuffin of Death and they can stop it before its an issue. But if they wait, they will have to deal with him and the McGuffin. Now, they have a real dilemma and rails won't help them.
To take it even further, highlight something unresolved or some goal on the character sheet. Point out that the world isn't standing still and any time they spend either helping this person or going after the big bad won't help them deal with that personal (to the character) matter. It too may get harder if they put it off.
Time is a great limiting factor in real life, it can be in games too.
Now, they have choices to make and no rails to help, they really have to choose. That is the first and biggest step.
After they get used to dealing with that, especially if you keep highlighting personal goals from the character sheet (and insist they add some if there aren't some obvious ones) they will eventually start adding more of their own, like becoming a dragon slayer. Then they will start actively trying to achieve those goals without further prodding.
Best Answer
I've played in and run evil campaigns of various sorts in both 3.5 and 4e (though not 5e, I think my learning will transfer), and run into a lot of problems: My Guy Syndrome comes up a lot, as does a tendency to default to a regular D&D storyline only with more stealing of spoons and kicking of puppies to remind ourselves we're evil. Sometimes an evil campaign instead descends into over-the-top motiveless violence until there's no story at all. There's a whole host of at-the-table and in-the-story issues, and I tried many different strategies to address them. Eventually I came up with a framing device which works well for us in avoiding these problems:
Provide the PCs with a Master to guide them toward orchestrated works of Evil.
Start the game with the PCs as underlings/minions/hirelings/apprentices/etc of a powerful evil NPC. The Master has a complicated Evil Plan and he tasks his minions to enact various parts as the Plan progresses: "Bring me the soul of a hound archon," "Raze the border keep," "Steal the Apocalypse Gem," "Help a spy infiltrate the paladin's ranks," and so forth, tailored to the PCs' abilities.
This provides the party a reason to work together despite having different agendas (and working together will hopefully bond them as friends so that they want to continue as a group) and establishes small achievable evil goals that accumulate into an Epic Evil Event.
All you need to do is ask the players to make sure their characters have a good reason to work for the Master: The serial killer likes having his rampages subsidised (and the Master protects him from the Law); the necromancer seeks to learn from the Master's experience and gain access to his libraries of forbidden lore; the mercenary's in it for the money and benefits.
Eventually the Apprentices will surpass their Master.
Expect the party to betray their Master at some point, hijacking his Evil Plot for their own gain: this is not only expected, but awesome. It's the Master's Evil Plot, not yours, and the story isn't about the Master--it's about his apprentices. Consider the Master to be training wheels for evil, setting an example which the party can then follow to surpass and overthrow their instructor as they level up.
This works because Evil Needs Goals.
As Ed describes so well and AgentPaper elaborates in the D&D context, evil needs concrete reasons motivating its actions. The Master provides goals and motives while the players find their feet in the new paradigm, channeling and guiding their exploration of what it means to be evil in ways compatible with the D&D paradigm without simply kicking puppies during a dungeoncrawl.
A word of warning: Alignment is tricky.
D&D has a history of the details and nature of alignment sparking major heartfelt arguments, because D&D alignments are not easily (or appropriately) matched to real-world philosophies and moralities; they're narrative simplifications to support the game's conceits and draw their power from storytelling conventions rather than from genuine moral complexity. Exactly what this means and how to deal with it are beyond the scope of this answer (and possibly this site, although there's a LOT of questions on the topic you can look at), but you should be aware it exists and be ready to talk with your players about what "Evil campaign" means to them so there aren't nasty surprises mid-game.