It seems Israel-firster wasn't invented by neo-Nazis. From an article by The New Leader from 1952, the oldest use of the term I could find: "I still remember the scorn with which, four years ago, the more vociferous among the "Israel-firsters" would react to the suggestion that America's pro-Israel stand might prejudice her strategic position in the Near East." Here's more information on The New Leader, which lets you know its history (the first to publish Aleksandr Solzhenitsyn in the US, the first to publish Martin Luther King, Jr.'s "Letter From Birmingham Jail".
A 1961 American Jewish Year Book article, quotes Abram Sachar, a prominent Jewish leader, denouncing "Israel Firsters whose chauvinism and arrogance find nothing relevant or viable in any area outside of Israel."
It seems to have been not uncommon, and there are several phrases such as Egypt-firsters, Britain-firster, China-firster, Europe-firster, Formosa-firsters, Earth-firsters, and yes, I even ran across one of Palestinian-firsters: " 'Palestinian Firsters' Are Wrong about Mid-East"
And to through a curve ball out there, the Central Conference on American Rabbis wondered in 1967:
"Would a Zionist be ordained by a faculty dominated by "American Judaism" firsters?"
The term was used at least intermittently until recently, even amongst Jews. A 1987 article from the Washington Post about a performance for high school students in Tel Aviv includes the line "Matt Cohen proudly proclaims himself an "Israel Firster". And a Feb 17, 1995 article from Forward (now The Jewish Daily Forward), entitled "Jewish Agency Gives Big Nod to Burg" says:
This seemingly Israeli-centered vision of the problems of the Jewish world may be a stumbling-block to Mr. Burg's acceptance on the part of American Jews, although many Diaspora Jewish leaders, wishing to remain anonymous, had whispered their support of him in conversations with the Forward. Yet Mr. Burg denies that he is an "Israel-firster."
(Note: I used ProQuest to find this issue, I couldn't find a link to it elsewhere)
It's worth pointing out that "[Country]-firsters" seems to mean different things in different contexts. Sometimes, it seems to mean that someone supports that country above others; at other times, it seems to imply that they want to focus on that country.
The phrase does show up sometimes in neo-Nazi rags, but they don't seem to have invented or popularized it, especially considering how far back the phrase goes, how often it's used by Jews themselves, and it's connection with other "[country]-firster" phrases. It seems that the extent of their connection was that they sometimes used this phrase that others used.
Saying someone is using language that comes from neo-Nazis is a pretty serious accusation, and should not be thrown around lightly. Also, keep in mind the way the accusations were used - not merely to mute criticism and discredit individuals, but to imply that critics of Israel tend to congregate with anti-Semites.
Often times these are smokescreens used to obfuscate the real issue. The goal is to throw around as many accusations as one can, whether they are true or not. At best, people will assume you are a person of integrity, and believe your accusations. At worst, you manage to cause your opponents to chase controversy after controversy, and by the time they prove one of your claims as false, you have three others that you are making.
Too many have taken these accusations as truth without realizing that the burden of proof falls upon the person making the claim. One hopes that the next time falsehoods are used in an attempt to police the discourse, we will remember the history of the people we are arguing with.