Well, considering the Enclave's advanced technology, I'm guessing they have some sort of method of staying in the air that isn't quite as obvious as the engine that's used in Mister Handies and Mister Gutsy's, at least in Fallout 3, those models have a noticable engine that keeps them in the air, now where the system is or how it works we don't know, but then again, it's science-fiction, we can't expect how all the futuristic technology works to be immediately obvious.
Also, why are EyeBots called EyeBots? They don't seem to SEE anything, but they do send pre-recorded messages. Wouldn't they be called RadioBots or something?
How can we be certain they don't see anything? We never get to see what the world looks like to an eyebot, and just because they don't have any sort of immediately obvious camera or similar setup doesn't mean they don't have any, they need to be able to avoid running into things and target their defensive lasers somehow, after all, though I suppose it's possible that they use a method other than optical location for this. And who knows, maybe President Eden uses them to observe what's going on in the wastes while having them spread the Enclave message at the same time.
I assume that they're mainly called that because they vaguely resemble a giant floating metal eye, though.