This is a great question and points to a place I should be more careful with my language. Thanks for implicitly pointing that out! In fact, no, I am not speaking primarily about the mentally ill, at least not in any generally diagnosable way. In referring to "nut cases" I have generally been speaking about people who want to be hermits because it validates a kind of strangeness and anti-social bent in them. Sometimes this phrase simply means these people are bizarre and feel that eremitical life is the same and thus gives them permission to remain as bizarre as they wish. They are not so much concerned with discerning a vocation which is divine in origin or edifying to others as they are seeking a way to enshrine and institutionalize their own personal mental and emotional idiosyncracies and eccentricities.
The next questions are also quite good and more difficult to answer. Mental illness comes in many different forms and degrees of control and stabilization. My general answer to the first part of your question is yes, some mentally ill persons COULD be hermits, but not all and not most. Regarding the second portion of the question, those that COULD be hermits are those whose illness is well-controlled with medication and whose physical solitude definitely contributes to their vocations to wholeness and emotional/mental well-being. There should be no doubt about this, and it should be clear to all who meet them. It should assist them in loving themselves, God, and others rather than detracting from this basic responsibility. In other words, solitude should be the context for these persons becoming more authentically human and maturing in that fundamental or foundational vocation for the whole of their lives. With this in mind I am thinking too that some forms of mental illness do not lend themselves to eremitical vocations: illnesses with thought disorders, delusions, hallucinations, fanatical or distorted religious ideation, and the like are probably not amenable to life as a hermit.
On the other hand, some forms of mental illness or disorder would (or rather, could) do quite well in an eremitical setting so long as the anachoresis (that is, the healthy withdrawal) required by the vocation is clearly different from that caused by the illness and does not contribute to it, but instead, even serves to heal it. Certain mood disorders, for instance, cause a defensive or reactive and unhealthy withdrawal, but it is not the same as the responsive anachoresis of the hermit. The person suffering from clinical depression who also wishes to be a hermit should be able to discern the difference between these two things and this requires a lot of insight and personal work. However, if a person suffers from clinical depression (or has done in the past) I would say it should be pretty well-controlled medically, and no longer debilitating or disabling before the person is allowed to make even temporary profession as a diocesan hermit. At the same time, provisions for adequate ongoing and emergent care and treatment should be written into this hermit's Rule of Life.
In any case, I think the decision to become a hermit when mental illness is a factor is something which requires the candidate and her spiritual director, psychiatrist or psychologist, and the diocesan staff to work together to discern the wisdom of. Mental illness per se should not always automatically preclude this vocational option, but there is no doubt that eremitical silence, solitude, prayer and penance can exacerbate rather than help with some forms of mental illness. Even in the completely healthy person eremitical solitude can lead to mental problems. Ordinarily we are made for a more normal type of communion or social interaction with others, and this is a particularly significant area for caution when dealing with mental illness. This is another place where some years spent as a lay hermit, especially under direction and regular and effective medical care, are especially helpful in discerning a vocation to eremitical life -- if initial permission to pursue such a thing is deemed wise at all. So, once again, thanks for your questions. They are quite good and, among other things, remind me to take greater care with language.