A flight changed to the next day is essentially the same as a long delay/cancellation as far as EU air passenger rights are concerned. But if you have been informed more than two weeks before departure, you only have a choice between going through with the alternative flights they are offering and cancelling your booking (in which case you would get your money back – even for a non-refundable fare – but they obviously would not have to transport you and you would have to make alternative plans yourself).
Financial compensation and additional support only apply if you are delayed on short notice (and especially if you are already en-route and, say, stuck at your layover point). If that's the case, under EU rules, the airline should both make sure you reach your destination, provide food and board as applicable and pay you some money for the delay (it's not even a refund, it might very well be higher than what you paid for your ticket - I have seen that with EasyJet for example).
Finally, even if you are not legally entitled to anything, it cannot hurt to ask. You might get something as a courtesy (perhaps more likely if you are a frequent flyer or travel in a premium class?) and at least one airline offers a complimentary hotel night or a tour of their hub city under some condition as a matter of course, even if there was no delay (see 12 hour stopover in Istanbul (Atatürk)).