No, it is not rude at all to say "thank you" when someone says "have a nice day" or something similar. In fact, saying "thank you" in response to well-wishes or kind words is a polite and courteous way to acknowledge the person's good intentions and show appreciation for their sentiment.
When someone wishes you well, such as saying "have a nice day," "take care," "good luck," or any other positive expression, responding with "thank you" is a natural and socially acceptable way to reciprocate the kindness. It conveys that you heard and appreciated their well-wishes.
Not saying "thank you" in such situations might come across as dismissive or ungrateful, so responding with gratitude is considered a socially appropriate and polite practice. It helps to foster positive interactions and shows that you value the person's goodwill.
Remember, expressing gratitude is a small act that can make a big difference in how people perceive you and your interactions with them. So, don't hesitate to say "thank you" when appropriate, especially in response to well-wishes and kind gestures.