The Oxford dictionary crowd seem to be in two minds about this. The OED lists eurozone uncapitalized, but with the note "also with capital initial". The Shorter Oxford and the online version both show it in all lower case. Some other dictionaries also show it with no capital letter.
And as pat notes, it's often written without a capital, so I've no doubt it's eligible to be a Chi word. But I wonder, should it actually be made a common word? I feel the word is pretty well known everywhere, although it's used a lot more frequently in the UK and Ireland than in other English-speaking countries. Any thoughts?