No. No, they won't.
Think about the distance to an airplane in flight. Next, think about the area of the sphere with that radius, and how little of that area even a giant telescope's mirror will intercept. You're not going to be blinded. You might be surprised, and you might lose your dark adaptation, but you're not going to be blinded.
Now, think about the area subtended by the telescope's field of view -- in the example above, a circle perhaps one minute of arc in diameter. What are the odds that a bright meteor will pass through that area during an observing session, slowly enough to linger long enough to cause damage? Pretty darn tiny. (Your odds may actually be worse if you aren't using the telescope, because then you're more vulnerable to a really bright meteor crossing your wide native field of view.)
The "real reason" eyepieces are rarely fitted to huge telescopes is because it's wasteful. The human eye is much less sensitive than the instruments normally fitted to such a telescope, and it doesn't record its perceptions for later analysis. Demand for real scientific observations from a large telescope always exceeds available time, so nobody wants to waste the machine's capability for some momentary sensual gratification.
As for safety, until planes are outfitted with multi-watt lasers specifically targeting telescope facilities, or until we're ambushed by a dense swarm of very bright meteors again targeting telescopes (so their apparent motion is slow enough to make them linger in a tiny field of view), our visual observers are pretty darn safe.
Besides, you only look through the eyepiece with one eye at a time, so you'll have a spare...