Yes - it looks like you need to run this as a multilevel regression model.

I have a slightly different take on this - the key is that the outcome variable (burnout or not) is not independent as workers are nested in work-groups and there may be an unexplained effect of belonging to a group which is driving this dependency. So I would use a two level model of worker in group and then you can put in the overall model the treatment variable ( it could be measured at the individual or group level ie clustered treatment model), measured attributes of the individual (your two dimensions), and any group level measured variables ( average of the dimensions for the group); the random effects will then be unexplained differences between groups, and unexplained differences within groups between individuals. I would still do this if there are singletons who do not work in a group; but you clearly have some groups,

You do not say much about the response - but if it is discrete, there are some tricky issues that need to be faced ( the nature of the level-1 variance; population average versus cluster specific estimates , calculating ICC when level-1 variance cannot go down when variables are included, obtaining good estimates when the number of workers in a group is small)). Good material on all of this can be found at

http://www.bristol.ac.uk/cmm/software/mlwin/mlwin-resources.html#discrete

I, too, have a different take on this. Generally, you want to go with the most parsimonious approach, and multilevel modeling (MLM) is generally not straight forward. Furthermore, for MLM, you particularly need sufficient power to make meaningful inferences. When you run ICCs, it is expected you would get some overlap as ICCs are correlations. The likelihood of non-meaningful correlations between ratings certainly depends on the design of the ratings, so this is something to take into account. I think it is as you put it. You have a potential moderator (the "when") of the relationship between treatment and outcome. If you find that the probability of the ICCs is high, then a moderated regression would make sense. I recommend reviewing your confidence intervals and not just the p-value. Let me give a silly but applicable example. If you asked everyone in the study to rate how much they enjoyed their breakfast at their site on a 3-point Likert scale, then you are going to have a lot of agreement if you analyze those data using ICCs, as there is likely to be little variance, and you would expect to see clustering based on site. So would that justify creating a fancy nested model based on breakfast ratings?

I definitely recommend looking into whether a moderated regression would be sufficient for your analyses, as MLM is a large undertaking if you're not already familiar.

Of course, Andrew Hayes's PROCESS macro for SPSS is a good way to go: http://www.processmacro.org/index.html

I think that there may well be unexplained variation between groups and there is a need to know this - it may be a bit more complex, but only a multilevel model will tell you this while conditioning on other variables. There is nothing very complex about the moderator in a mixed model - it is basically the same as in standard regression, that is an interaction. But you will need sufficient power to be able to detect effects, see this site for how to do power calculations bespoke fro your problem

http://www.bristol.ac.uk/cmm/software/mlwin/mlwin-resources.html#datadesign