you are unlikely to get sick if you take both shots at the same time. they are totally different vaccines with different side-effects and immunological responses. i often give (killed) flu vaccine and (live) chicken pox vaccine on the same day to children. if those vaccines are free to you and you do not have any contraindications to receiving them then i do not see a problem. both diseases are miserable so since they can be prevented then go for it.
is it wise to get shingles vaccine and flue vaccine at the same time? . my work is offering a shingles vaccine and flu vaccines. they will pay for them. but can i get them both on the same day? would i get really sick? . . . .