My answer is a bit long, so I'm going give you the short answer first.
Virtue is a necessary condition of achieving eudaimonia.
However, this is different than the answer to the question as it was formulated at the end of your post, "Is there an argument that a good life can be lived without being good?". One could easily argue that a person can live a good life, independently of virtue, simply by providing examples of individuals who have a "good life" but who are not virtuous. However, if you are using the phrase "living a good life" to explicitly mean "achieve eudaimonia", then an individual cannot achieve eudaimonia without virtue.
As you seem to be aware, there are multiple conceptions of eudaimonia. The three best documented views are those of the Stoics, Artistole and Epicurus. I will quickly summarize their views as relevant to your question:
the Stoics: Virtue is necessary and sufficient to achieve eudaimonia. The Stoics believed that eudaimonia cannot be achieved without virtue.
Aristotle: Aristotle addresses your question in Book I of the Nicomachean Ethics. You can read the complete text at http://classics.mit.edu/Aristotle/nicomachaen.1.i.html . In summary, Aristotle believes that virtue is necessary, but not sufficient because "good fortune" is an additional requirement for achieving eudaimonia. Just like the Stoics, Aristotle believed that eudaimonia cannot be achieved without virtue.
Epicurus: Epicureanism is the closest we will come to a eudaimonia that can be achieved without virtue, but even his Hedonistic philosophy requires virtue to achieve eudaimonia. According to Epicurus, eudaimonia is achieved by successfully pursuing and maximizing pleasure. The catch is that the only way to successfully pursue and maximize one's pleasure is to practice virtue. In Epicureanism, virtue is nothing more than a means to an end, but it is the only possible means, so it is still a necessary condition of eudaimonia.
Below, I have included two passages from a paper I wrote in 2008. These passages contain a much more in depth explanation of Aristotelian eudaimonism and Epicureanism, as well as documentation and reference to primary sources.
Excerpt from On the Compatibility of Kantian Ethics and Aristotelian Eudaimonism
This content is fully copyright protected. The first excerpt is from the section explaining Aristotle's eudaimonism. The unspecified citations that appear refer to Terrence Irwin and Gail Fine's translation of Nicomachean Ethics. The second excerpt is a discussion of Epicureanism in contrast to Aristotelian eudaimonism.
On Aristotelian Eudaimonism
Eudaimonism is the set of ethical theories that consider eudaimonia to be the highest good and advocate its pursuit as right moral action. Eudaimonia is a Greek word, which translates most closely to happiness. However, "happiness" is an insufficient and incomplete characterization of the concept. Aristotle says that "most people virtually agree about what the good is, since both the many and cultivated call it happiness, and suppose that living well and doing well are the same as being happy. But they disagree about what happiness is, and the many do not give the same answer as the wise." Clearly, there is disagreement among eudaimonist theories as to what exactly eudaimonia is. The main points to understand are that eudaimonia is the word used to describe the highest good in all eudaimonist theories, that there are various conflicting conceptions of eudaimonia and that "happiness" is the English word to which most often eudaimonia is translated. The focus of this discussion will now shift to a discussion of Aristotle's conception of eudaimonia.
Aristotle views eudaimonia as the activity of "living well" and "doing well". It is not a mental state of satisfaction that one achieves by such activities, but the conscious activity itself. Eudaimonia has the characteristic of being the good at which all other goods aim. He calls eudaimonia "the highest of all realizable goods" and asserts that "the best of all things must...be something final" (I, 4). All other good things, such as "honour and pleasure and reason, and all virtues or excellence, we choose partly indeed for themselves...but partly also for the sake of happiness, supposing that they will help to make us happy" (I, 7). Aristotle contrasts seeking these goods with seeking eudaimonia, which "no one chooses... as a means to anything else at all" (I, 7). So, all other goods are, at least in part, a means to achieving eudaimonia. Therefore, eudaimonia is a "complete end", and not a means to any other end. The other goods, which are both ends in themselves and means for achieving eudaimonia, can be called virtues.
Aristotle investigates eudaimonia further with the "Function Argument". Basically, he asserts that every thing has an inbuilt function or purpose for existence, and excellence of a thing is defined by excellence in its purpose. For example, a television has the inbuilt purpose of entertaining its owner with television shows. An excellent television will entertain its owner with the television shows that he wants to see, in High-Def without ever losing picture. Things can be used for purposes other than their inbuilt purpose. For example, I use my television as a surface on which to display empty liquor bottles. However, it is the inbuilt function that Aristotle speaks about.
He asserts that just as everything else has an inbuilt function, human beings must as well. Furthermore, he explains that the inbuilt function of a person must be related to a characteristic of humans that make them specifically human. He writes "we take the human function to be a certain kind of life, and take this life to be the soul's activity and actions that express reason" (I, 7). To Aristotle, humans are unique in that they are rational beings with a faculty for reason. Since excellence of a thing requires the excellent performance of the function by the thing, Aristotle determines that an excellent human will reason excellently and act in accordance with excellence of reason. The sign of a human completing his function well is that he "expresses the proper virtue". So, "the human good turns out to be the soul's activity that expresses virtue."
Aristotle also requires that an agent experience some good fortune in order to obtain eudaimonia. Because eudaimonia is a constant activity, there will be both good fortune and bad fortune in the life of an agent. The fortune relevant to eudaimonia is that which is present when you look at the big picture of the agent's life (I, 9). So, Aristotelian eudaimonia is the highest realizable good. It is an activity of the soul that must be performed over time, and that depends on exercising excellence in reason through virtue. Also, eudaimonia requires sufficient good fortune of the agent that he may recognize his own well being.
On Epicureanism
We will now turn to an examination of the moral philosophy of Epicurus. Epicureanism is also a type of eudaimonism, but is very different than that of Aristotle. Epicurus asserts that "pleasure is the starting-point and goal of living blessedly" (Epicurus. Letter to Menoecus. 128). Epicureanism is based on empirical observation. Cicero explains this when he writes "as soon as each animal is born it seeks pleasure and rejoices in it as the highest good, and rejects pain as the greatest bad thing" (Cicero. On Goals. 1.30). For Epicurus, successfully pursuing and maximizing pleasure is necessary in order to achieve eudaimonia.
Epicurus does not think that pleasure alone constitutes eudaimonia. It is also important that pain is avoided. Many pleasures, such as "drinking bouts and continuous partying" and "consuming fish and the other dainties of an extravagant table", can have negative effects (Epicurus. Letter to Menoecus. 129). Epicurus does not think people should pursue pleasures that are accompanied by pain. Instead, people should maximize their constant exposure pleasure by being virtuous. Epicurean eudaimonia is a life of pleasure while virtue is a means to obtain it.
This contrasts with Aristotelian eudaimonia in the relationships between eudaimonia, virtue and reason. To Epicurus, virtue is nothing more than a means to obtain eudaimonia. It has no intrinsic value as an ends. However, Aristotle holds that virtue is good as a means to obtaining eudaimonia but also good as en ends in itself. Furthermore, Aristotle's conception of virtue relies on exercising excellence of reason, and is justified by his Function Argument. While Epicurus uses reason to evaluate empirical observations, he does not make the same connection between virtue and reason as Aristotle.
