It seems the OR statement in the Implicit region is not behaving as expected. As far as I can see plotting a single implicit region works. The solution of Hans allows for plotting them both, however you want something that works more general correct?
One option would be to wrap everything in a function that generates the code Hans provided.
r1 = eq1 && x == 0 && y == 0;
r2 = eq1 && eq2;
dom = {x, y, z};
RegionsPlot[eq_, dom_, opts : OptionsPattern[]] :=
RegionsPlot[eq, dom, Automatic, opts]
RegionsPlot[eq_, dom_, rng_, opts : OptionsPattern[]] :=
Show[ImplRegions[eq, dom, rng], opts,
(*default plot options*)
Axes -> True, AxesOrigin -> {0, 0, 0},
AxesLabel -> {"x", "y", "z"},
AxesStyle -> Directive[Medium, Black, Bold]
];
ImplRegions[eq_, dom_, rng_] /; ListQ[eq] :=
ImplRegions[#, dom, rng] & /@ eq
ImplRegions[eq_, dom_, rng_] /; Head[eq] == Style :=
Block[{eqp, style},
{eqp, style} = eq /. Style[e_, st___] -> {e, {st}};
Region[Style[ImplicitRegion[eqp, dom], style,
(*Default region styling*)
PointSize@Large, Thick
], PlotRange -> rng]
]
ImplRegions[eq_, dom_, rng_] := ImplRegions[Style[eq], dom, rng]
Then this should work, with the function propagating all options given to Show.
RegionsPlot[r2, dom, rng]
RegionsPlot[Style[r1, Green], dom, rng]
RegionsPlot[{Style[r2, Red], Style[r1, Blue]}, dom, rng]