learning experience
Much of the feedback so far weighs in on the side of "required -- family is family". Someone cited honoring one's parents (the source of the request), and a couple people mentioned protecting a life (the sibling is apparently in real danger of injury without someone there).
I, on the other hand, am leaning toward "forbidden", though "permitted" is a possibility. Definitely not "required", though.
The issue is complex. While the sibling needs a caregiver, that's a service that can be hired -- so there's no apparent need for the poster to do it personally. Of course it's important to honor one's parents (this comes up a lot in text), but the talmud also teaches that if a parent asks you to transgress the Torah, you must decline (Bava Metzia 32a). This raises the question of whether attending another religion's worship service -- on its second-holiest day, to boot -- is avodah zara, forbidden worship. Is it enough if you don't intend to worship? What if you don't participate? What if you don't listen? That is a complex question with varied answers depending on circumstances, ranging from exactly what will take place to the strength of your own Jewish education and commitment, and you really need to ask your rabbi for a personal ruling.
I think the experience of facing this issue is valuable for the conversion candidate, actually. As a member of a minority religion (that sometimes faces hostility from others), sometimes you are going to have to make choices between your religion and your family/friends/society -- things like this, or resolving Shabbat issues with your employer, or various other matters. Finding out how you will handle those choices before it's "too late" -- before you convert and acquire new obligations -- seems useful to me.
I assume that most conversion candidates face some sort of religion-vs-world-at-large test during the process, but I don't actually know.

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Now my story is of going with my wife to Christmas eve services. I had just settled into the Chinese food when a passing harmful remark she made to her sister, as the latter was going out the door, prompted her to want to go to the church and make up with her sister. I've been to Christian services before, but the whole lighting the candle thing and all made me distinctly uncomfortable. But my wife found it valuable, because she felt so alienated being there, so much that it just was not her, that she left a stronger Jew than she had been before.
We talk about the Christmas thing (ie being at her house with the tree and all) every year, because I really hate Christmas, yet it is very important to her family, and I know she feels the tug back to them. The stakes are higher with the baby, too. And there's the fact that her family is pretty amazingly anti-Semitic. But I have decided that it's part of her and I know that she is not a complete person when she feels estranged from them. Darchei shalom is a Jewish value too, so we live with it for now. That's how I would advise the person you originally referred to, to think about it.
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In a sense it seems kind of extremist to me... yet at the same time I can see their point. Although, to be honest, I think the specifically Catholic obsession with saints, and even more so with the "Virgin Mary", is far closer to idolatry.
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But even more so, I'm not sure I do see their point. "Avodah zarah" originally meant offerings, human and otherwise, to false gods. Christians differ on the nature of God, but they are monotheistic. I also think that in a sense it's a disservice to not recognize a faith that, for all the bad history there, did spread the idea of monotheism around the world.
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I forget which branch of Christianity (the one without the icons and most of the above), bt it's ok for a Jew to enter their church because their practice of Christianity doesn't fall under idol worship
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However, I'd not point to genuflecting as a problem; we have our own version, after all. And, unfortunately, one can stretch a point (possibly too far) to claim that we've invested perhaps a bit more iconography in Torah scrolls than is justified by the need to show proper respect for them. (On the other hand, we're not perfect either; maybe we need to do some self-examination on that issue.)
I will agree that the whole business with the cross is unjustifiable iconography, as is the "fish" that is increasingly common, and other icons used by various denominations (many Protestant branches use an icon of flame with a cross motif worked in, and then there's the Catholic "sacred heart").
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Also, afaik, Greek Orthodox is the Eastern church that ignores Rome, and Byzantine Catholic is the Eastern church that follows Rome. It would seem that the former should be the less influenced of the two, being, I'd guess, made up of those broken-off factions you mentioned, if they still exist. (It seems to make no sense that the Romans would just let half the Eastern church leave if they had managed to take them over when the Byzantine Empire fell.)
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As a consequence, whenever I go into a certain kind of church (for example, on the Freedom Trail in Boston), I think, "gee, this looks just like a shul!")
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So my point was that it may or may not be avodah zara, but in this case it's unnecessary for the person to land in that situation, and in general you should discuss the topic with your rabbi. The answer may well be "no problem; go", but you need to check and not assume, IMO.
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Yes, it seems to me (and I am not Jewish, though I am Ashkenazi) the question is not whether it is permitted/forbidden/required for a Jew to do this, but whether it is permitted/forbidden/required for this Jew to do this.
Well, that's one question. But (being practical as is my wont) another, less theological one is: has anyone checked with the handicapped sister to inquire how she might feel to be saddled with a keeper who was pointedly not joining in with the service? Mightn't she feel uncomfortable with that situation?
At any rate, I feel the need to point out the ludicrousness of someone needing a Jew to accompany them to a Christmas service. We're talking about a gathering of a whole barn full of Christians. I can't believe there isn't a single one there who can be convinced to take a sister-in-faith under her wing for one night. What, does she think nobody else is going to show up? Her fellow congregationists, what are they, chopped liver?
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Good point!