Because $N$ is an odd square, it must end with $1$, $5$, or $9$. Meanwhile, the following relation must also hold: $$N \equiv 1\pmod{4}$$ However, it is easy to see that none of $11$, $15$, $19$, $31$, $\cdots$ satisfies this property, but all of $21$, $25$, $\cdots$ do. Hence, we can conclude this claim hold.

(Note: $1$, $5$, $9$ differs by $4$, and $10$, $30$, $50$, $\cdots$ differs by $20$. Both of them are multiples of $4$. Therefore, we just need to check the first term $11$. If it does not satisfy $N\equiv 1\pmod{4}$, none of them will do.)