**EDIT**: My previous answer was completely wrong.  Here's a corrected version. 

First, note that this property descends to subgroups: every irrep of a subgroup is a summand of a restriction from the bigger group. Thus if all the irreps of the bigger group are 1-d, all the irreps of the small group will be their restrictions with the same one coming up multiple times.  

If $G$ has $n$ 1-dimensional quaternionic representations whose sum is faithful, then $G$ must embed into the product of $n$ number of copies of the unit quaterions (that is, $SU(2)$).  Consider the image of $G$ in any one of these representations, and then consider the image in $SO(3)$; since it is a finite subgroup of $SO(3)$, it is either cyclic, dihedral (including $C_2\times C_2$), or $A_4,S_4$ or $A_5$ (see [Wikipedia](http://en.wikipedia.org/wiki/ADE_classification)).  However, if it's nonabelian, this group has a real irrep (an irrep over the real numbers whose endomorphisms are $\mathbb{R}$, which is thus absolutely irreducible) which isn't 1-d, so it tensoring gives an irrep over the quaternions that isn't 1-d, so it must be cyclic or $C_2\times C_2$.  This shows that the original subgroup of $SU(2)$ is either abelian, or the classic quaternion group.

Thus, $G$ is a subgroup of a product of cyclic and quaternion groups, and as we argued above, all such subgroups have this property.