I've come across a solution which works. It basically uses a phantom block to raycast from Player to any Eyeball in line with the Player, and if that block hits an Eyeball before it hits a Wall, it will make the Eyeball move toward the Player:
[Player | Eyeball] -> [Player | < Eyeball]
[Player | no Wall |... | EyeBall] -> [ Player | > Gaze | ... | Eyeball]
[> Gaze | no Wall | ... | Eyeball ] -> [ | > Gaze | ... | Eyeball ]
+ [> Gaze | Eyeball ] -> [ | < Eyeball > Gaze]
[ Gaze ] -> []
This works fine even for multiple Eyeballs, apart form some slight hiccups when Eyeballs compete for a square (which Eyeball wins is probably decided by the internals of the engine at resolution of movement, and would be tricky to solve using rules because doing so would either require to match patterns that are not along a single line, or to move the Eyeballs manually using absolute directions).
That is to say, if you have two Eyeballs both touching the Player, and the Player moves out of the way, only one Eyeball will move to fill in the spot left by the Player, and which one that turns out to be is decided by the engine when the movement occurs.
(This competition would probably be tricky to solve using rules because doing so would require to match patterns that are not along a single line.)
If there is only one Eyeball, then the first two rules may be put together and simplified to
[Player | ... | EyeBall] -> [ Player > Gaze | ... | Eyeball]
This runs into issues with more than one Eyeball because several Gaze blocks cannot coexist in the same cell, meaning only one Gaze block ever gets made. One could also mend this by putting again after each rule, but that would make moves not happen instantaneously so it's not optimal.