Wolfram Programming Lab gives the slightly "simpler" $$ \sum_{k=n}^{\infty} \frac{(k!)^2}{(k+x)(k-n)!(k+n+1)!} \\ = \frac{(n!)^2}{(n+x)(2 n+1)!}\ {}_3F_2(n+1,n+1,n+x; 2n+2,n+x+1;1), $$ with a hypergeometric function instead of Meijer's G-function.

Wolfram Programming Lab gives the slightly "simpler" $$ \sum_{k=n}^{\infty} \frac{(k!)^2}{(k+x)(k-n)!(k+n+1)!} \\ = \frac{(n!)^2}{(n+x)(2 n+1)!}\ {}_3F_2(n+1,n+1,n+x; 2n+2,n+x+1;1), $$ with a generalized hypergeometric function instead of Meijer's G-function.

Edit: In case $x$ is a negative integer and $x \le -n$ ($n$ is assumed to be a non-negative integer) the hypergeometric function is Saalschützian (see MathWorld) and thus we get: $$ {}_3F_2(n+1,n+1,n+x; 2n+2,n+x+1;1) =(-1)^{n+x} \Gamma(1-(n+x)) \frac{((n+1)_{|n+x|})^2}{(2 n + 2)_{|n+x|}} $$ with $(a)_k$ the Pochhammer symbol.