The finger nails peeling off could be because of various reasons like change in the climate,nutritional deficiency
,some medications which he was on , sometimes post viral infection like in hand Foot mouth disease
and fungal infection
of the nails.
At present vomiting
and fever could be just a viral infection and the finger nails have nothing to do with the present illness.He needs to continue with the medication and if not better in a couple of days then evaluation by his docter.Get his finger nails issue be assessed by a dermatologist