Despite the growing application of interactive technologies like service robots in customer service, there is limited understanding about how customers respond to interactions with frontline service robots compared to those with frontline human employees. Moreover, it is unclear whether all customers respond to the interaction with frontline service robots in the same way. Our research looks at how individual differences in social behaviors, specifically in customers' attachment styles, influence three types of customer responses: affective responses (experienced pleasantness), attitudinal responses (perceived empathy, satisfaction), and behavioral responses (word‐of‐mouth). Three experimental studies reveal that customers with low (vs. high) scores on anxious attachment style (AAS) measures respond more negatively to frontline service robot (compared to a frontline human agent). We investigate alternative explanations for these findings, such as robots' level of anthropomorphism and we show that human‐likeness features such as voice type and level of human‐like physical appearance, cannot explain our findings. Our results indicate that for low‐AAS customers replacing frontline human service agent with frontline robot undermines customer attitude and behavioral responses to service robots, leading to possible implications on customer segmentation, targeting, and marketing communication.