I can’t give you a source but I am not so sure that Chazal said lice do not come from eggs. I believe they said lice do spontaneously arise from the earth or decaying matter or whatever it is. I don’t believe they excluded other forms of genesis as well. This is how I recall it.
Or for that matter they may have considered “eggs” to be a normal part of the complex components of dirt, as we consider spores, bacteria, fungi, and decaying organic matter to be.
I also am not so sure “lice” is the correct translation of the organism that is being referred to.