This is not an answer and I cannot comment due to lack of reputation, but it seems to first pop up in his Lemma 4 on p. 50. This occurs after a long, very technical discussion about double coset operators. The goal is to show that two specific representations of the Hecke algebra are equivalent (one whose image lies in $End_{\mathbb{C}}(S_{k+1/2}(N, \chi))$ and the other in $End_{\mathbb{C}}(S_{2k}(N))$). The Lemma:
Lemma 4 (p. 50). Every elliptic or hyperbolic conjugacy class in $\Gamma_{0}(4N)C_{n}\Gamma_{0}(4N)$ contains an element $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $d > 0$, $(b, d) = 1$, and $\left( \frac{b}{f}, \frac{t^{2} - 64n^{2}}{f^{2}} \right) = 1$. Here, $t := a + d$ and $f := (d-a, b, c)$.
Now, for squarefree $M$ set
$$\mu(t, f, n, M) := \prod_{p \mid (M, f)} (1+p)\cdot \#\{x \in \mathbb{Z} : 1 \leq x \leq M, (x, M) = 1, x^{2} - tx + n \equiv 0 \mod (fM, M^{2})\},$$
Lemma 5. Let $A \in GL_{2}(\mathbb{Z})$ be an elliptic or hyperbolic matrix with $\det A = 16n^{2}$, $t \equiv 0 \mod{4}$ and $f$ odd. Then for $N$ odd and squarefree there are $\mu(\frac{t}{4}, f, n^{2}, N)$ matrices $B \in \Gamma(1)/\Gamma_{0}(4N)$ with $B^{-1}AB \in \Gamma_{0}(4N)C_{n}\Gamma_{0}(4N)$.
So it seems that the definition of $\mu$ depends on $M$ being squarefree, and in turn this allows Kohnen the counting argument in Lemma 5. I don't know why squarefree-ness is needed from this, though. But maybe this information can help you or someone else determine the answer.
EDIT:
Answer: If $N$ is not squarefree, there exists a case such that all common eigen-subspaces of $S_{k+1/2}^{+}(4N, \chi)$ for Hecke operators have dimension at least 2, whence a strong multiplicity one theorem does not hold (see Ueda's On twisting operators and newforms of half-integral weight).
We may try to decompose our cusp space into eigen-subspaces of twisting operators (why??). Let $$M := \prod_{\substack{p \mid N \\ ord_{p}(N) \geq 2}} p^{ord_{p}(N)}, \quad \Pi := \{p \text{ prime} : p \mid M\}.$$ One can decompose the Kohnen space as
$$S_{k+1/2}^{+}(4N, \chi) = \left( \bigoplus_{\kappa: \Pi \rightarrow \{\pm 1\}} S^{0, \kappa} \right) \oplus Ker\left(R_{\Pi}; S_{k+1/2}^{+}(4N, \chi) \right),$$
where $S^{0, \kappa} := \{f \in S_{k+1/2}^{+}(4N, \chi) : f \mid R_{l} = \kappa(l)f, \forall l \in \Pi\}$ and $R_{\Pi}, R_{l}$ are the twisting operators of the characters $\prod_{l \in \Pi} \left( \frac{*}{l} \right)$ resp. $\left( \frac{*}{l} \right)$. The 'kernel' part consists of oldforms, and each $S^{0, \kappa}$ is stable under the Hecke operators $T(n^{2})$ for $(n, 4N) = 1$.
Why we need squarefree: There exists a case such that $S^{0, \kappa} \cong S^{0, \kappa'}$ as Hecke modules for distinct $\kappa, \kappa': \Pi \rightarrow \{\pm 1\}$. Hence when $M \neq 1$ Kohnen's theory does not work.