Such a value $d$ does not necessarily exist. For $J=(xy,xz)$ we have $(xyz)^{d-1} \notin J^d$ for any $d$. It’s even worse (but maybe a little degenerate) if $J$ is principal.
A necessary and sufficient condition for existence of such a $d$ is that the monomials generating $J$ don't have a common factor. To be explicit:
Proposition: Let $m_1,\dotsc,m_k \in R = \Bbbk[x_1,\dotsc,x_n]$ be square-free monomials, $J=(m_1,\dotsc,m_k)$. (We don't assume $k=n$, or that the monomials have the same degree.) There exists a $d$ such that $(x_1 \dotsm x_n)^{d-1} \in J^d$ if and only if for each $i=1,\dotsc,n$, there is (at least) one of the monomials $m_j$ such that $x_i$ does not appear in $m_j$.
Proof: As in the example above, if each $m_j$ is divisible by the same $x_i$ for some $i$, then every generator of $J^d$ is divisible by $x_i^d$, so $(x_1\dotsm x_n)^{d-1} \notin J^d$. Conversely, if the condition is met, then in the product of the generators $m_1 \dotsm m_k$, each variable $x_i$ appears at most $k-1$ times, so $m_1 \dotsm m_k$ divides $(x_1 \dotsm x_n)^{k-1}$, and $(x_1\dotsm x_n)^{k-1} \in J^k$, i.e., $d=k$ (the number of monomial generators) works. $\square$
For a square-free monomial $m$, let $\overline{m}$ be the set of variables that don't appear in $m$. So the condition above is that $\overline{m}_1 \cup \dotsb \cup \overline{m}_k = \{x_1,\dotsc,x_n\}$.
Proposition: With notation as above, $(x_1 \dotsm x_n)^{d-1} \in J^d$ if and only if there exist $j_1,\dotsc,j_d$ such that $\overline{m}_{j_1} \cup \dotsb \cup \overline{m}_{j_d} = \{x_1,\dotsc,x_n\}$.
Indeed in the product $m_{j_1} \dotsm m_{j_d}$ each variable is omitted from at least one factor, hence appears at most $d-1$ times. Conversely if $(x_1 \dotsm x_n)^{d-1}$ is divisible by a generator of $J^d$, say, $m_{j_1} \dotsm m_{j_d}$ (here there is no assumption of distinctness of factors), then each $x_i$ must be omitted from at least one of the $m_{j_\ell}$'s.
So the minimum $d$ that works is the same as the smallest size of a cover of the set of variables by the $\overline{m}$'s. I'm not a combinatorialist but it seems to me that such $d$ cannot be determined only from the sizes of the $\overline{m}$s and pairwise intersections. It must involve triple intersections, etc. If you only want a bound then I suppose: the size of $\overline{m}_1 \cup \dotsb \cup \overline{m}_d$ is greater than or equal to the sum of the sizes of the $\overline{m}$'s minus the sum of the sizes of pairwise intersections (baby Bonferroni inequality). So if $d$ works, every $m$ has degree $e$, and each two $m$'s have at least $f$ factors in common, then the pairwise intersections of $\overline{m}$s have size at least $n-2e+f$, so
$$
n \geq d(n-e) - \binom{d}{2}(n-2e+f),
$$
and you can solve this quadratic inequality for $d$ and see if that gives an at all useful bound. I hope it helps.
(Edit: The pairwise unions of $\overline{m}$s have size at most $n-f$, but I should have bounded the size of the pairwise intersections. Again, sorry any confusion.)
