Legal Challenges
You're asking if there is any way for the Illinois GOP to prevent Arthur Jones from being the Republican candidate for the US House.
Yes. They can challenge the validity of the signatures he collected to qualify for the ballot or accuse him of other violations of the election code.
In fact, in a previous attempt for the nomination, he was removed from the ballot for invalid signatures and other legal violations. (He's attempted to get the nomination six times since 1998.)
As of this writing, however, it doesn't appear that there is anything else that can be done. There have been no challenges to his petitions. And it's too late to run challengers, even as write-ins. The filing deadlines have passed.
Why?
But a larger question is why would we want to prevent a candidate from running?
In a free country, anybody who meets the constitutional requirements should be allowed to run for Congress.
Political parties, or any other group for that matter, should not be in the business of suppressing free expression, regardless of how detestable it may be.
Political parties should encourage a free exchange of ideas and welcome all candidates into the arena. Let them fight it out. May the best man or woman win.
Arthur Jones got the required number of signatures on his petitions to qualify for the ballot. Preventing him from running disenfranchises the citizens who support him.
In the US we should not prevent a candidate from running. The right way to beat a candidate is to run your own candidate. The Illinois GOP failed to do that. If a Nazi gets the nomination, they only have themselves to blame.
Context
Due to gerrymandering throughout the country, most seats in the US House, and many seats in state legislatures, rarely, if ever, switch between parties.
The re-election rate in the US Congress over the past 50 years is consistently 85%+, with the rate reaching 98% in many election cycles.
In my home state, New York, the re-election rate in the state legislature is always 90%+.
From the New York Times:
Last year, more than half of the 212 legislators in the Senate and Assembly won with more than 80 percent of the vote. Fifty-seven ran unopposed, according to the New York Public Interest Research Group. The average senator has served for nearly seven two-year terms.
So, basically, until there is redistricting reform on a national scale, a huge number of legislators can enjoy job security nearing what's given to a federal judge (lifetime appointments). They just need to watch out for primary challengers. (Sadly, this creates a class of high priority constituents made up of party leaders and activists, while the general population can be safely ignored.)
That said, most congressional districts in the US go without a serious challenge. This creates a vacuum. It can be filled by anybody. The third congressional district of Illinois is a good example. The seat has been solidly Democrat since 1975. In 2016, the Democrat won with 100% of the vote. In 2014, the Democrat received 65% of the vote. The Illinois GOP doesn't bother investing anything into this race. With no standard bearer, a wide assortment of characters see a realistic pathway to the nomination.
However, with a news media that is generally anti-Republican and anti-conservative, and always on the look-out for a sensational story, a GOP candidate who could be portrayed as a Nazi (regardless of their odds of winning) was bound to make national news.
