Is this Spanish correct? Ustedes chicos no debe tener una competencia para contar mentiras,” dice.?

It's enough to write: "Ustedes" instead of "Ustedes chicos", which is not correct at all. The verb "deber" (to must in this case) is not correctly conjugated, for the plural of second person, it is conjugated as "deben", but if you want to say "shouldn't", the correct conjugation is "deberían" (another tense). Then "you shouldn't"="(ustedes) no deberían" and you mustn't="ustedes no deben".

"Competición" is correct, but "competencia" is more coloquial.

Actions in Spanish never are said with present continuous, so "mintiendo" is not correct. The correct is to say "de contar" ("not para contar", "para"=for). Actions are designated by the infinitive or by nouns, never by present continuous (gerundio).

The correct sentence is "Ustedes no deben (deberían) tener una competencia de decir mentiras", and that means "You mustn't (souldn't) have a lying competition (or a competition to tell lies)".

I don't think "chicos" is necessary. Ustedes implies that you're speaking to a group of people. Also, when asking or exclaiming anything, you need to have the reverse version of the question mark or exclamation point at the beginning of the sentence.

Ex.:

¿Cómo estás?

¡Bien!

Like that. On a Windows computer ¿ is alt+0191 and ¡ is alt+0161. Hope this helps.

"These boys don't need to have a competency in order to tell lies"--that is what your spanish sentence above says.

NOW, if you are speaking to the boys, you would leave out the 'chicos' and say "You don't need to have. . ."

Ustedes no ____________________ (change the conjugation of 'debe' to fit ustedes)