Have a nice time.

[Précis. "Mijte se hezky!" does not translate into English as "Have a nice/good time," but you can use this sentence in your leave-taking if you know what the other person will be doing. "Have a nice time" always has a reference to something specific you know about the other person's plans because s/he told you. The English equivalent of "Mijte se hezky!" is probably something like "Enjoy life."]

Two people meet on the street. They know each other. They chat about nothing for a few minutes--small talk, you know. They part and one says to the other, "Have a nice time." What does the person who says this know about the other person?

A. Something specific about what the other person will do next or sometime later.
B. Nothing, just saying goodbye and wishing the other person well.

The correct answer is A, Something. The one who says "Have a nice time" knows what the other person will do next, or where s/he is going, or what his or her plans are. In other words, "Have a nice time" has a reference to something known by both people. Always.

So when you want to say goodbye to someone, do not say "Have a nice time" unless you know something specific about what will happen with the other person. "Have a nice time" is not a substitute for "Have a nice day" or "See you later," etc. It is a substitute for goodbye only if you both know something about what's next for one person.

Can one say "You too!" in reply? Sure, if the other person knows something specific about your near-future activity.

The misuse of this farewell is so common in the Czech Republic that foreigners who have lived here for some time probably just ignore its misuse, which doesn't make it correct.

How can one learn this fine point of English usage? First, the Internet will not help you except that you will not find "Have a nice day" as a farewell or leave-taking expression. Examples of different expressions of this type appear here.

http://en.wikipedia.org/wiki/Parting_phrase#Generic_ways_to_say_goodbye

And here.

http://www.fluentu.com/english/blog/say-goodbye-english/

If you do find "Have a nice day" as a recommended way of saying goodbye, you are probably looking at a web site or forum where non-native speakers are giving other non-native speakers advice, or their guesses as to what is proper English usage and what is not. (Example, https://dewisudjia.wordpress.com/2010/12/22/expressions-of-leave-taking/.)

But, you object, even Radio Prague (http://old.radio.cz/en/html/living_pozdravy.html), a trusted source for cultural and linguistic bits, says "many Czechs say 'Mijte se!', or its informal equivalent 'Mij se!', which means 'take care'. An nicer way of saying this is 'Mijte se hezky!', which means 'have a good time!'" And the resident interpreter in my house says there is no referent, just a general "enjoy whatever."

So here is the answer for the prevalence of the misuse of "Have a nice time." It is likely a matter of translating a language and culture directly into English without knowing how English speakers would say the same thing Czechs want to when using Czech.

Now you know your homework: Find a way in English to say the same thing you do in Czech. Have a nice time (doing that!).

Postscript. Here is a two-word scramble answer, one of many possible answers.

Eefijlnoy