Almost half of adults between 16 and 24 said they had experienced stress or anxiety, compared to just over a third of all UK adults.
Young adults were also more likely to be uncomfortable talking about a mental health problem, with one in three saying this compared to 27 per cent of all adults.
13 per cent also said they were experiencing a problem but had not sought help, compared to seven per cent of all adults.
The firm's medical expert, Dr Doug Wright, said the term could cause problems. "Our findings suggest that young adults are more likely to be experiencing mental health problems, so using a phrase which criticises this age group could add to this issue.
"Any term used disparagingly to a segment of the population is inherently negative.
"While young adults in particular appear to take offence to the 'snowflake' label, the majority of adults agree that the term is unfair and unhelpful, so it's important that people consider how such labels are used, and the cumulative effect they could have on their recipients," he said.
The phrase, which originated in America, has particularly come to refer to universities and students who use "trigger warnings" to alert readers or listeners to potentially distressing material, and "safe spaces", which are meant to be free of certain opinions or ideas.
It is also used to imply that young people are less resilient than older generations and quicker to take offence.
It is thought to originate with US author Chuck Palahniuk's 1996 book Fight Club, which contains the line "You are not special. You are not a beautiful and unique snowflake".
In an interview earlier this year he took credit for the term, adding: "My friends who teach in high school tell me that their students are very easily offended."