Regarding Kennedy specifically, there were concerns that he would effectively be taking orders from the Pope. He addressed these concerns himself in a speech in 1960, in which he rejected that premise:

But let me stress again that these are my views—for contrary to common newspaper usage, I am not the Catholic candidate for President. I am the Democratic Party's candidate for President who happens also to be a Catholic. I do not speak for my church on public matters—and the church does not speak for me.

Here is the full text of his speech given on September 12, 1960. This speech was given in Houston, Texas. A number of other speeches are available from his library, but I don't know exactly which one you are referring to.

Lyman Beecher, a leading 19th-century Protestant minister and father of Harriet Beecher Stowe, offers his reasoning in an 1835 text.

He argues broadly that Catholicism is incompatible with republicanism (that is, representative government). His contention is that Catholics take direction from their priesthood closely and the priesthood in turn takes direction from the Pope. A democracy of Catholics would not be following the self-determined will of the people so much as the will of the Pope. Church and state are united under Catholicism, he contends.

The Pope of the time, Gregory XVI, in his own way agreed with Beecher. In this encyclical he asserts the divine right of kings, opposes freedom of speech, and advocates for the 'mutual concord' of church and state.

Both Beecher and Pope Gregory XVI claim that republican democracy and Catholicism are antithetical to each other.

Of course, that's just part of it. In the 19th-century, particularly from the 1830s onward, Catholicism intersected with other important statuses. Namely, Catholics in the US were also mainly immigrants. Hundreds of thousands of Catholics were immigrating into a largely Protestant country. They were seen as threatening to the status quo, creating a confluence of nativism and anti-Catholicism. Resistance to Catholic immigration notably took the form of the Know Nothing movement. The strength of that movement eventually faltered in the face of more salient issues (e.g., slavery and the Civil War) but suspicions of Catholics' anti-democratic nature lingered long into the 20th century.

It's not that Christmas was too Brittish, the Puritans practice Christianity according to the word of God. There is no word of God even describing a season, much less a date for the birth of Jesus. Only shepherds watching a flock on a hill in Judea, which is NOT something you would be doing in winter. Combine this with the fact that the Church of Rome set that date near the, already celebrated, Pagan mid-winter festival in the 4th century and you have a holiday that the Catholics invented and the Puritans hated. So Christmas was too Catholic, not too British.

Maybe not the best source but there is a really great article in this months "New England Antiques Journal" that has a lot of early American Christmas information, titled "Christmas Banned in New England -by John Fiske"

here: https://www.antiquesjournal.com/flipbooks/neajdec14/files/34.html