1800s usage, probably from the imagery that eggshells are easily broken when stepped upon.
Pronunciation
Lua error in Module:parameters at line 290: Parameter 1 should be a valid language or etymology language code; the value "Received Pronunciation" is not valid. See WT:LOL and WT:LOL/E.IPA(key): /ˌwɔːk ɒn ˈɛɡʃɛlz/
Lua error in Module:parameters at line 290: Parameter 1 should be a valid language or etymology language code; the value "General American" is not valid. See WT:LOL and WT:LOL/E.IPA(key): /ˌwɔk ɑn ˈɛɡʃɛlz/, /ˌwɑk-/, /-ɔn-/
(idiomatic) To be overly careful in dealing with a person or situation because they get angry or offended very easily; to try very hard not to upset someone or something.
(idiomatic) To be careful and sensitive, in handling very sensitive matters.