Should I expect worse video quality with GoPro using lens protectors?
If you use lens-protection of any kind, same with autocue or eyedirect, the additional glass in front of your camera will have some effect, yes. The magnitude of which will vary depending on your uses. Autocue glass usually has a slightly murky / milky look to it and the image might seem a little but soft upon close inspection - I assume the same will go for GoPro lens-protectors, since they are made of similar transparent material. The absolute worst that could happen is to get your protector dirty. If you have splotches of water or dirt on that protector, they will be very visible and distracting.

tl;dr: If you keep the lens-protector clean, you should experience very little effect. If absolute sharpness is a must for you (when you're filming with green-screen for example), then I would not recommend using it.