No, consider the following counterexample: Take $$ `M = \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix}` , \quad J = \begin{pmatrix} 1 & 2\end{pmatrix},$$$$ M = \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix} , \quad J = \begin{pmatrix} 1 & 2\end{pmatrix},$$ then $J^# = `\begin{pmatrix} 1 \\ 0\end{pmatrix}$$J^\# = \begin{pmatrix} 1 \\ 0\end{pmatrix}$ and your projection is given by $I - J^# J = `\begin{pmatrix} 0 & -2\\ 0 & 1\end{pmatrix}$$I - J^\# J = \begin{pmatrix} 0 & -2\\ 0 & 1\end{pmatrix}$ and this is definitely not non-negative definite by your definition.
Edit: Can't seem to get the matrices right...
