The Eurozone isn't a separate entity from the European Union, but an umbrella term for the countries that have reached the third stage of the Economic and Monetary Union. Technically all member states of the EU that have met the convergence criteria are required to adopt the Euro, with the notable exceptions of Denmark and the United Kingdom who managed to obtain special opt outs in the 1992 Treaty of Maastricht.
According to Article 50 (3) of the Consolidated version of the Treaty on European Union, if a country withdraws from the European Union all treaties regarding the EMU, starting with the Treaty of Maastricht, will no longer apply, which effectively means that a Eurozone member withdrawing from the EU will also be withdrawing from the Eurozone:
3. The Treaties shall cease to apply to the State in question from the date of entry into force of the withdrawal agreement or, failing that, two years after the notification referred to in paragraph 2, unless the European Council, in agreement with the Member State concerned, unanimously decides to extend this period.
That said, the previous paragraph allows for the withdrawing member to negotiate its future relationship with the EU, without excluding the possibility of the withdrawing member continuing being part of the EMU or any other specific treaty or sets of treaties. Thus, it is theoretically possible for the withdrawing member to remain within the Eurozone, although it seems highly improbable that the European Parliament will give its consent, especially since 2004 opting out of the third stage of the EMU is not allowed for past or future EU members.
Lastly, there is nothing legally stopping a country that has withdrawn from the EU to adopt or continue using the Euro as its national currency, similarly to how Monaco, San Marino, the Vatican City, Kosovo, Montenegro and Andorra have adopted the Euro without being members of the EU and without being represented in the Euro Group. This seems like a far more likely possibility, assuming of course that the withdrawing member is interested in adopting or continuing using the Euro.
The question borders on the meaningless because leaving the EU would be such a dramatic process that it would require complex negotiation and a set of new treaties. As others already discussed, there is in fact an article about that in the treaties but it seems very theoretical at this point.
So what's currently in there doesn't seem very relevant, it's the politics that would matter, more than ever. If there is a will, there is a way, if you will (although I am not too sure there would be a will in this case).