Are mindfulness and flow related Loose definitions for mindfulness and flow (text from Wikipedia):

Mindfulness involves being aware moment-to-moment, of one’s subjective conscious experience from a first-person perspective.

Flow is described as a deep focus on nothing but the activity – not even oneself or one's emotions.

Both are good, but they appear to be opposite mental states. Is entering a flow state incompatible with developing mindfulness? If so, then would flow be considered "bad"? If not, then how can these distinct concepts be presented so that they are compatible?
Equating mindfulness and flow is somewhat problematic.  I would actually go so far as to say that flow has more in common with samadhi than sati.

I'm guessing you are using the term flow as described by Mihaly Csikszentmihalyi.  From what I remember of his book, one of the important characteristic of flow inducing activities is that they have an engaging quality to them.  One becomes absorbed in whatever they are doing.  That should set off some bells and whistles immediately as there are some clear similarities (but not an equivalence) between being engaged in an activity and states of meditative absorption.  For one, both activities are, by nature, intrinsically rewarding (well, when you get better at meditation, they are!) and offer direct feedback.  We know when we've had a good sit or not, we know when our mind drifts from the the breath, etc.  On a more technical level, attention is applied and sustained (vicara/vitakka) in both flow and samadhi.  We either need a meditative object to apply our attention to (samadhi) or an activity to which we can apply our efforts (flow).  We also need to do both in a way that isn't distracted (sustained attention).  In flow, however, I just don't think one finds a sustained attention that is nearly as unbroken as it is in jhana or neighboring states of concentration.  

When compared to mindfulness practice, flow is lacking some very important qualities.  For one, I just flat out don't think the word mindfulness even can be applied to flow.  The Pali word sati has a number of meanings, but generally implies bringing something into the mind's eye, bringing something into attention, etc.  I just don't see that taking place in flow.  The activity is supplying the mental content.  Secondly, mindfulness practice involves sampajanna - clear comprehension - of the activities one is engaged in.  This is the knowing that a breath is long or short, that an arm is raised, that a feeling is felt, etc.  Once again, I just don't see that happening in flow.  One is caught up in the activity instead of clearing comprehending the component actions that make up that activity.  Perhaps most importantly, flow utterly lacks the potential for insight.  Flow never affords one the opportunity to turn the attention to, say, the seven factors or enlightenment or the four noble truths.  The activities take us into a kind of attention that is other or outer-directed.  The potential for the development of wisdom - the main purpose of mindfulness practice - never factors in.