The answer to your question is "no". Let $G=GL(2,\mathbb C) \times GL(2,\mathbb C)$ act on $X=GL(2, \mathbb C)$ by $(A,B)\cdot C= ACB^{-1}.$ The orbit of the matrix $x=\left(
  \begin{array}{cc}
    1 & 1 \\\
    0 & 1 \\\
  \end{array}
\right)$ is $X$. (Hence it is closed.) Let $H$ be the subgroup of matrices of the form
$\left(
  \begin{array}{cc}
    a & 0 \\\
    0 & a^{-1} \\\
  \end{array}
\right) \times \left(
  \begin{array}{cc}
    a & 0 \\\
    0 & a^{-1} \\\
  \end{array}
\right)\subset G$. Then $Hx=\left(\begin{array}{cc}
    1 & b \\\
    0 & 1 \\\
  \end{array}
\right),$ for $b\ne 0.$ Hence $Hx$ is not closed.