Yes, hypochondria can cause itching. If you have hypochondria and you think you have a rash, then you will worry about the rash and have anxiety about the potential negative consequences. So the anxiety and fear of having a rash can induce adrenergic pruritis, or stress-induced itching. Beta blockers can block the effect of adrenaline, and can effectively reduce the itch (Haustein, 1990).
Reference:
Adrenergic urticaria and adrenergic pruritus. Haustein UF. Acta Derm Venereol. 1990;70(1):82-4.