Do gargoyle geckos have eyelids

Nope - if they need to 'clean' their eyeballs - they use their tongue to lick the surface.

Crested geckos have a clear membrane covering the eye, this membrane is a clear, transparent eye contact more or less. it stops dirt, oxygen, water, substrates etc.. from entering the eye. when something external touches the membrane, they will lick their eye in order to remove anything. The Crested geckos come from the Diplodactylinae subfamily under the Genera Rhacodactylus. Geckos of the Eublepharinae Subfamily are known as "eye lid" geckos all have eye lids. These consist of the: Leopard gecko (E. macularius), the fat tailed gecko (E. Hemitheconyx) etc.. When sleeping, resting, swallowing, and of course blink, their eyelids will close.