We have no jobs sounds more like something someone would say if they were informing someone else that their establishment is not hiring.
We don't have jobs is more of a statement about the speakers themselves, rather than an establishment with which they are affiliated.
"Do you have any jobs going?" -> "Unfortunately, we have no jobs at the moment." (Our company is not currently hiring)
"Where do you guys work?" -> "We actually don't have jobs right now." (We are currently unemployed)
In short, the translation you suggested can be taken as more informal and more descriptive of the person who is speaking, whereas Duolingo's translation can be taken as something someone would say if the company has no positions available and someone had asked about it.
But, as far as I'm aware, there is no distinction like this in German (basically, nicht is used to negate a verb, kein is used to negate a noun), so technically either translation should work.
I would also like to make it clear that this distinction may not follow in all English-speaking places. I can only comment on how I perceive it as a native speaker from my part of the USA (Maine). So don't assume that this is a steadfast rule in English. To some natives, they may be completely interchangeable.
Es tut mir Leid. Obwohl ich seit eine Weile Deutsch lerne, ist mein Deutsch viel zu schlecht, das alle auf Deutsch zu schreiben. Das Thema ist zu schwer, mein Vokabular ist zu klein, und meine deutsche Grammatik ist zu schlecht aber ich hoffe, dass es zumindest ein bisschen Hilfreich war.