>>>It doesn't actually run out until October this year but we are going to Dubrovnik in June and the passport will be invalid, I believe
Croatia joined the EU on the 1st July last year. By law, all EU member states are obliged to recognise all EU passports throughout the FULL validity stated on them. (You could enter Croatia, using your existing passport, with only a few hours validity left on it. Since, as an EU citizen, you're entitled to remain there indefinitely, there can be no restriction on you using your passport to gain admission but, obviously, it helps if it's valid when leaving as well).
From the website of the Foreign & Commonwealth Office:
"Your passport should be valid for the proposed duration of your stay.
No additional period of validity beyond this is required."
https://www.gov.uk/foreign-travel-advice/croatia/entry-requirements