The following is what Mazur wrote on page 91 of his paper, Modular curves and the Eisenstein ideal, published in Publ. IHES in 1977, DOI: 10.1007/BF02684339 (freely available at eudml):

Let $m$ be an integer. Let $J[m]_{/\mathbb{Z}}$ denote the scheme-theoretic kernel of multiplication by $m$ in the Neron model of $J_{/\mathbb{Z}}$. **Since $J$ is semi-stable, $J[m]_{/\mathbb{Z}}$ is a quasi-finite flat group scheme**, whose restriction to $S'=\text{Spec} \mathbb{Z}[1/N]$ is finite and flat.

Here, $J=J_0(N)$ with $N$ prime.

I have two questions:

It seems that his claim is still true when $N$ is squarefree because $J$ is still semi-stable. But I don't know why. Can someone explain this?

Suppose that $N$ is not squarefree, so $J$ does not have semistable reduction at a prime whose square divides $N$. Is the same statement true if $\text{gcd}(m, N)$ is squarefree? If not, is it true that if we assume that $\text{gcd}(m, N)=1$?