American Heritage Dictionary:
"Eskimo has come under strong attack in recent years for its supposed offensiveness, and many Americans today either avoid this term or feel uneasy using it. It is widely known that Inuit, a term of ethnic pride, offers an acceptable alternative, but it is less well understood that Inuit cannot substitute for Eskimo in all cases, being restricted in usage to the Inuit-speaking peoples of Arctic Canada and parts of Greenland. In Alaska and Arctic Siberia, where Inuit is not spoken, the comparable terms are Inupiaq and Yupik, neither of which has gained as wide a currency in English as Inuit. While use of these terms is often preferable when speaking of the appropriate linguistic group, none of them can be used of the Eskimoan peoples as a whole; the only inclusive term remains Eskimo. •The claim that Eskimo is offensive is based primarily on a popular but disputed etymology tracing its origin to an Abenaki word meaning “eaters of raw meat.” Though modern linguists speculate that the term actually derives from a Montagnais word referring to the manner of lacing a snowshoe, the matter remains undecided, and meanwhile many English speakers have learned to perceive Eskimo as a derogatory term invented by unfriendly outsiders in scornful reference to their neighbors' unsophisticated eating habits."
http://www.yaelf.com/aueFAQ/mifeskimo.shtml is somewhat less sparing of people's sensibilities, which is refreshing (to be honest). I particularly like that the Inuit decided that the Yupik should be called Inuit, if that's a reasonable inference from what is said there.
http://en.wikipedia.org/wiki/Eskimo also seems reasonable and compatible with yaelf, while adding new information. The American Heritage's 'disputed' is probably to be read as 'almost certainly false', its 'the matter remains undecided' equates roughly to 'not capable of being proven beyond all doubt'.
So it's a question of whether we avoid using a term one group erroneously finds objectionable in favor of using a term of dubious application to other groups. There was a spate of this kind of thing in the '60s (give or take 20 years), with groups finding the most objectionable interpretation or origin of a non-self-assigned name, and propagandizing it. But the most objectionable interpretation was frequently not the most likely, or nor in some cases even a possible, origin. Hence the entire 'squaw' boondoggle. Even worse, speakers are frequently unaware of the original meaning of the ethnonym, anyway: I liked Am. Her.'s 'learned to perceive'.