Glawariel,
My mistrust is aimed at any candidate who seems to be subservient to interests other than the civic interests of our own country.
I threw in the State of Israel, meaning the Knesset, distinct from the Rabbinate, as indication that it is not only regilious affiliation that might compromise a candidate's position in my eyes. The Jewish neocons who 'represent' Israeli interest within our current administration alarm me just as much as the Christian Fundamentalist contingent.
Because I am an American and not an Israeli, I feel strongly that my support for Israel, in whose survival I hold a stake as a Jew, should be expressed through the WZO and not by attempting to elect candidates within the United States who would place Israeli interest above American interest. Endorsement by the Likud (for example) is not necessarily indication of this - I doubt I could define any one thing that would be by itself an indication of this, but I submit that I would know it when I see it, and I would not vote for such a person.
It is treacherous to use Israel as an example of this - to point a finger at this potential - because 'dual loyalty' has long been one of the charges used to justify persecution of Jews. That was the charge against Kennedy when he ran as well.
And I do not think the concerns about a Catholic president were necessarily eyewash, because the Vatican has no small amount of political and economic power, and it may be that Kennedy's several anti-Communist boondoggles did arise from the 'indoctination' of his Catholic upbringing. There was talk at the time that his support for the Diem government in Vietnam was at the urging of the Vatican, but my own opinion (just from cursory knowledge, not from analysis) is that Kennedy did not pursue specific actions because of undue influence but rather that he probably thought a certain way about global issues because of his Catholicism, and of course there is no way to avoid the general influence of background on any candidate. Rational people do not vote on this basis alone. The question is whether the candidate is transparent about their policy for America and whether its main object is service to America. As much as I disagreed with Kennedy on many issues, I do not believe that he held any dual loyalties.
I dared to use Israel as an example, in spite of the historic imprudence of doing so, in order to use my own People as an example and not someone else. But I have the same feelings about the role Cuban Americans have played in US politics. I would not vote for a Cuban American whose support for anti-Castro foreign policy was a centerpiece of their campaign, and I would not vote for a Jewish American whose support for Israel was a centerpiece of their campaign. It was wildly distressing to me when we learned that a Catholic Bishop was withholding communion from Catholic candidates who did not oppose abortion. This crosses a line for me. And if Secretary Wolfewitz views the US Defense machine as an aegis for the Israeli defense machine, that crosses the line for me as well.
We live in an area of Philadelphia that is sort of an old Tory haven (the Main Line) ... very old English money here. And before concerts and sporting events they sing God Save the Queen. I have to tell you that this creeps me out.
It also troubles me to sing Hatikva in the synagogue. There's something about the national anthem of another country that does not feel to me appropriate as a
public expression within the United States. Find something else to express ethnic affiliation, but not a national anthem.
This is not to say that I would give my vote to someone who advocated the abandonment of Israel! But a
particular policy on Israel - using Knesset endorsement as a sort of barometer - would not be a deciding issue for me at the polls. I have other civic mechanisms available to me for protecting my civil and human rights as a Jew within America, and for exercising whatever influence in Israel I might feel entitled to as a Jew, i.e through the WZO. It is also my opinion that using American administrations to clear a path for Israel has not been good for Israel. Crossing one's objectives and loyalties never has a happy ending, in my opinion, no matter what the issue is. If your first loyalty is to Israel or to England or to Cuba, then go live in Israel or England or Cuba.
It's unfashionable to look at things like race, religion, ethnicity, or even foreign affiliation when forming political opinions. How well I know! But it is impractical to ignore them, imo,
IF they appear to have significant influence on the candidate's opinions.
And we are talking here about my one vote - on what bases might I withhold that from a candidate. Anything that caused me to suspect they raised some other value above their civic service to America would cause me to withhold my vote from them.
Jn
many people who have some sort of belief in an afterlife, and because they are good people, they are not, I know, the sort of people who would make callous decisions if they were to find themselves In Charge! And the people who do do such things are probably fairly callous generally, and would go to war and order the destruction of others even if they had been born without the faith gene. 
We sing other of our national songs with far more enthusiasm. Strangely, we have no particular worry about other anthems being played. The Marseillaise and the Star Spangled Banner as the best known ones would be treated with respect and anyone complaining would be seen as a bit unusual. You might notice that our Last Night of the Proms party concert which is about as nationalistic as the English get has scores of foreign flags waved merrily in the audience.