How dear ! Bruised egos or ignorance or old age again .
Back to the book : declaring an emergency will give you priority and assistance from ATC. Period . It is not an admission of failure ,
Squawking 7700 will just get you priority and, as already said here , will warn everyone else that you might divert from your flight plan ,and/or altitude and that is including the military ,
in todays world of pancakes sectors there are plenty of other sectors below you , some even located in another country .
So when ATC ask you to set 7700 , if is to make sure you will not hit someone else on the way down and possibly to save you the cost of being intercepted .
As to refusing to do it , of course as PIC you always can refuse to follow any ATC instruction , you just will have to explain why afterwards in an office over tea/coffee.!