There's truly no bad time of year for your photos in Chicago. Spring, summer, fall and winter all have an abundance of uniquely beautiful scenery to offer. Think about what seasons and time of year you enjoy best.
Maybe you love snow and chic winter coats... or maybe beach weather and boating is your calling. Consider what weather and temperatures you'll feel most comfortable taking your engagement photos.
As for day of the week: At this time, I typically only schedule engagement sessions Mondays through Thursdays. This is largely because my calendar is reserved for weddings which occur Fridays, Saturdays and Sundays.