Thanks for your questions. Without knowing the person specifically it is hard to say why someone might say they are careful not to tell folks they are a hermit because of their response to a life of solitude. We hermits, whether lay or consecrated, are what we are, and eremitical life is (or should be!) a healthy way to fullness of life for us; we tend to be open about the fact that we are hermits and that this is the way God calls us to wholeness and fullness of life. People may have a hard time understanding this because of how unusual it is to be called to authentic humanity through the silence of solitude, but in my experience anyway, so long as they can see the essential truth of this claim we make, they won't or don't tend to react badly. And maybe that's the key to what is at the heart of the caution spoken of by the person you are asking about: somehow the solitude of eremitical life does not seem to truly contribute to the person's wholeness, well-being, and fullness of life. As your questions regarding my own life seem to suppose, some chronic illnesses might not allow one to live as a hermit -- not even as privately vowed.
I don't think folks have greater difficulty understanding how it is I live in solitude than they would of anyone else. Some are concerned for me and how I manage during times of illness or injury but once they realize I am free to ask for assistance should I need it (indeed I can say I am morally bound to do so!) their concerns are eased. Additionally, my relationship with my Director (delegate) and relationships with folks in my parish faith community help assure that I have the assistance I need to get to doctor's visits or to consider various options for treatment, hospitalization, etc. But all of this considered, what is more important I think, is that folks tend to see I thrive in solitude and are assured that I work with superiors, et al., in order to be sure that continues to be true. Yes, I live in solitude and a medically and surgically intractable seizure disorder can make that problematical in some ways, but I am genuinely happy as a diocesan hermit, and the canonical structures of my life along with other relationships make it relatively easy to reassure people that the pros of eremitical solitude far outweigh the cons. Again, as I have noted here before, because this vocation is ecclesial it is not a form of isolation but of significant solitude defined in terms of community and that is something folks may not understand. When these things are borne in mind I do not find it difficult to reassure folks that eremitical life is demanding but incredibly rewarding, nor that though it is uncommon, it is not abnormal or unhealthy for one truly called to it.
There are certain dimensions of my life, certain times of suffering and struggle, which are integral to what I consider to be the way God has shaped the "heart of a hermit" over my 70 years of life; these are indeed difficult to share with others. I hold them private and share them with very few people. But God has always been there for and with me and this graced part of events comes through in the theology I teach or write, in the very limited preaching I do and the Scriptures I most resonate with and teach. When I speak or write about eremitical life I always stress the redemptive element which is at its center, and in fact, must be at its center if it is to be considered of God. There is no reason usually to write about the various dimensions of my own suffering, for instance, unless doing so really serves to illustrate the degree or quality of God's grace in my life at the same time. My life is fruitful and richly meaningful because of the love of God that has accompanied me in every moment and mood of that life. The God I believe in, the Creator God and Abba of Jesus Christ, is not one who wills suffering; instead he wills life and this means he wills to accompany each of us in our suffering not only that life and meaning can be drawn out of what would otherwise merely be death dealing but so that one day all suffering is ended and God will be all in all.
All of this and more, by the way, is what the Church examines when she seeks to discern an eremitical vocation with someone. The story the Church seeks to hear is the story of God's redemption of a person as this occurs in the silence of solitude. A bishop and his Vicar for Religious, among others, will be listening for the grace that dominates and makes sense of all things as a person who seeks to become a diocesan hermit enters into the mutual discernment process required for admission to the consecrated state. In other words what makes my own life understandable and reassures others about the healthiness and fruitfulness of eremitical solitude is an expression of the very same thing Church hierarchy listens for if they are to accept a person has a genuine eremitical call to life under canon 603. For that matter the hermit's Rule of Life will also provide evidence of this same narrative. Similarly, if a person cannot move successfully through such a discernment process it might also be the case that they will refrain from telling others they are a hermit, not only because it is difficult to admit such failure, but because they may not actually be able to reassure folks sufficiently that eremitical life is really a healthy or fruitful choice for them.
Public profession will commit one to witnessing to eremitical life as a way to a fruitful, healthy life which sings of God's grace and strikes others as being happy. Should health demands or other life circumstances move the hermit away from being able to witness in this way in spite of the suffering involved the hermit may be required to consider seeking or accepting a dispensation of her vows. Still, while the vows are binding a person may well be bound to the elements of canon 603 and eremitical life others do not "get". It is important to be clear these vows are made freely and can, if necessary, be dispensed if the calling is no longer truly healthy for one. Meanwhile, if one's embrace of eremitical solitude is a matter of an entirely private commitment (private vows), one is always obligated to keep the superseding values of their public baptismal state. Such private vows will not, generally speaking, include any commitment to eremitical life per se nor any obligation to live under an eremitical Rule, and they may well reflect an inadequate discernment process in any case. A private commitment to eremitical life may well need to be left behind if the life proves unhealthy for the person whether or not private vows of the evangelical counsels also need to be dispensed --- something easily done by one's pastor, in every case.