When an X spell is on the stack, the value of X contributes to the CMC. In every other zone, Chalice has a CMC of 0, but on the stack, if X = 2, the CMC will be 4 (2 + 2). If they declare X = 0 or 1, the CMC is 0 and 2 respectively and those are the only ways it will trigger Eidolon. If X is 2 or greater, Eidolon will not trigger.
It's 4, so Eidolon won't trigger. The value of each X is counted in the converted mana cost of the spell while it is on the stack. Everywhere but on the stack, X is undefined and translates to 0 for CMC calculations, but as the card is a spell, we have a value to work with.
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I'm a former judge (lapsed), who keeps up to date on rules and policy. Keep in mind that judges' answers aren't necessarily more valid than those of people who aren't judges; what matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.
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Eidolon vs XX