If you're an introvert, why do you have a job that involves dealing with people all day to begin with? It sounds to me like it isn't lunch that's the problem, it's the hours before and after it.
It's not an all or nothing thing. It's not as if interacting with humans is physically painful. Being introverted isn't a crippling social disorder; it just means that sometimes you need to recharge by spending time by yourself, and often find that relaxing.
(I actually do work at home now, but I've worked in office environments in the past, which I've liked fine, not least because I got to spend lunchtime when, where, and with whom I chose.)
(I actually do work at home now, but I've worked in office environments in the past, which I've liked fine, not least because I got to spend lunchtime when, where, and with whom I chose.)