  In this paper squared eigenfunction symmetry of the differential-difference
modified Kadomtsev-Petviashvili (D$\Delta$mKP) hierarchy and its constraint are
considered. Under the constraint, the Lax triplets of the D$\Delta$mKP
hierarchy, together with their adjoint forms, give rise to the positive
relativistic Toda (R-Toda) hierarchy. An invertible transformation is given to
connect the positive and negative R-Toda hierarchies. The positive R-Toda
hierarchy is reduced to the differential-difference Burgers hierarchy. We also
consider another D$\Delta$mKP hierarchy and show that its squared eigenfunction
symmetry constraint gives rise to the Volterra hierarchy. In addition, we
revisit the Ragnisco-Tu hierarchy which is a squared eigenfunction symmetry
constraint of the differential-difference Kadomtsev-Petviashvili (D$\Delta$KP)
system. It was thought the Ragnisco-Tu hierarchy does not exist one-field
reduction, but here we find an one-field reduction to reduce the hierarchy to
the Volterra hierarchy. Besides, the differential-difference Burgers hierarchy
are also investigated in Appendix. A multi-dimensionally consistent 3-point
discrete Burgers equation is given.
