Perfect squares that are also predecessors or successors of two-digit numbers are $9$, $16$, $25$, $36$, $49$, $64$, $81$ and $100$. Two-digit numbers that have a perfect square as their successors or predecessors are therefore $10$, $15$, $17$, $24$, $26$, $35$, $37$, $48$, $50$, $63$, $65$, $80$, $82$ and $99$. But a two-digit number that is preceded or succeeded by a prime must be even, so the only options are $10$, $24$, $26$, $48$, $50$, $80$ and $82$. Out of these, the only ones with a prime neighbour are $10$, $24$, $48$, $80$ and $82$. All five of these satisfy the condition. Thus, the correct answer is $5$.