Not in a case like this. It's the content of what was said, that's important. In this case it was a warning that the act could kill someone. That statement is what's to be evaluated, not the claim of authority. In general, claims of authority are meaningless, only what is said matters. The truth of what is said is not determined by authority in any event. Only trust is extended.
The DJs didn't address the claim of being a nurse. They addressed the matter that the action could result in death. They took on the role of authority and indicated they knew someone might die and indicated, that they didn't care.
I understand the point, but you're missing the issue. DJs get calls from people from all walks of life. Some are the genuine article, some aren't - they don't know whom to believe and cannot be held liable for the content of the information.
I can rell you that I'm a climatologist and that all the fossil fuels that are being burned to keep your computer on the Internet are contributing to Global Warming, but it doesn't make any of it true. You can choose to believe that GW is caused by substances such as fossil fuels, or greenhouse gases, or cows farting in New Zealand. The fact is that, depending on your beliefs, it doesn't make any of it true. A court might allow one of the attorneys to introduce this as "evidence", but it doesn't make any of it binding on either the DJs or the station.
However, had there been a nurse in the studio during the contest, the whole story would be substantially different. But, as far as we know, that was not the case.
By the way, I'm the Surgeon-General of the United States. Don't smoke and practice safe sex and . . . . . stuff like that.