Does Green's $\mathcal{J}$-relation define a total order on the equivalence classes? In a semigroup $S$ we define $a\le_\mathcal{J} b$ iff $a=xby$ for some $x,y\in S^1$. Defining $a\equiv_\mathcal{J} b$ by $a\le_\mathcal{J} b$ and $b\le_\mathcal{J} a$ gives a partial order on $S/\equiv_\mathcal{J}$. Is this order always total?

No it is not always a total order. 
For example, in the semigroup $S$ with elements $a$, $b$, and $c$ and multiplication in the table below:
\begin{equation*}
\begin{array}{c|ccc}
&a&b&c\\\hline
a&a&a&a\\
b&a&b&a\\
c&a&a&c
\end{array}
\end{equation*}
The $\mathscr{J}$-order on $S$ has $b$ and $c$ incomparable, $a\leq_{\mathscr{J}} b$ and $a\leq_{\mathscr{J}} c$. 
More generally, every partially ordered set with a minimum element is order-isomorphic to the partial order of the $\mathscr{J}$-classes of some semigroup. A reference for this is Theorem 4(i) in the paper 

C. J. Ash and T. E. Hall, Inverse semigroups on graphs, Semigroup Forum 11 (1975)
  140–145.

