The unique continuation is valid for  generalized Laplacians. The   operators you mentioned are such generalized Laplacians.  This follows from Hörmander's result in 

> *Uniqueness theorems for second order elliptic differential equations*, Comm. Partial differential Equations, 8(1983), 21-64.


Suppose that $E$ is a metric vector bundle ver the Riemann manifold $(M,g)$. Denote by $\Delta_M$ the  scalar Laplacian determined by the metric $g$. Fix a connection on $E$ compatibvle with the metric on $E$ and set $\Delta_E=\nabla^*\nabla: C^\infty(E)\to C^\infty(E)$. 

For any $u\in C^\infty(E)$ we have

$$\Delta_M |u(x)|_E^2=2\big\langle \Delta_E u(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

Any generalized Laplacian   $L:C^\infty(E)\to C^\infty(E)$  then there exists a conection $\nabla$ on $E$ compatible with the metric on $E$  such that 

$$ L=\nabla^*\nabla+ T=\Delta_E+T,$$  

where $T$ is an endomorphism of the bundle  $E$; see [Proposition 10.1.34 here][1].

If $Lu=0$ then $\Delta_E u=-Tu$ and we deduce 

 $$\Delta_M |u(x)|_E^2=-2\big\langle Tu(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

At this point you can invoke the above results of Hörmander for the scalar function $|u(x)|_E^2$ to obtian the unique continuation.


  [1]: https://www3.nd.edu/~lnicolae/Lectures.pdf