The problem I found with eyelids is they need to move in an arc, whereas bones cause mostly linear motion in a mesh, so unless your eyes are flat, the lids are likely to intersect them when closing.
My solution, which is applied to human eyes with spherical contours, was to use two Shrinkwrap modifiers to conform the mesh to the surface of the eyeball while the bones move them. The first Shrinkwrap (I call it Eye Socket) makes the eye socket fit around the eyeball, flush to the surface, as my eyeballs are nearly complete spheres. The sockets are hidden for the most part but it helps keep things tidy as the eye rotates & the lids blink. There is some influence from Eye Socket also on the inner part of the visible eyelid, but it does not extend further. The Offset Distance for this Shrinkwrap is 0 (zero), and the mesh sticks right to the eyeball’s surface where the corresponding vertex group is weighted at 1.0. The eyelid vertices have lesser weights.
The Eye Lid Shrinkwrap keeps the outer surface of the eyelid riding over the eyeball but at a greater Offset distance, about 3-4mm in my models (their scale is 1BU = 1m). The vertex weighting for this SW modifier is usually much less than the other, rarely over 0.50 if I recall correctly, as there are other influences on this part of the mesh from the eyelid bones.
With proper balancing of the Shrinkwraps that keep the mesh conformed to the eyeball in two layers (inner & outer), I then use a pair of bones to Scale the mesh of each eyelid on only the vertical axis. One bone affects the upper lid, the other adjusts the lower. A single control bone operates them both. On my models the lids close differentially, the upper closing more than the lower. Normally scaling like this would result in a linear closing of the eyelids, but the Shrinkwrap keeps the lids mesh wrapped around the eyeball, so I can ignore that issue.
As usual, careful vertex weighting is necessary, but Weight Painting these vertices is inconvenient so I assign weights in the Object Data>Vertex Groups panel to selected edge loops that ring the eyeball.
This may not be a viable solution for you as your eyeballs don’t look truly Spherical, but maybe it will spark an idea or two that don’t depend only on using armature bones.