frank s is exactly right with the explanation for the pink carnation; Don Mclean is trying to say he had no date, probably for what we know as "prom" here in the States...
As for the 'broncin' buck', his definition is correct. In the context of the song, I believe Don is trying to say that he was a lonely teenage boy who, like so many of us at that age, could be very difficult to to get to know and spend time with, and so be difficult to be friends with... and that it would be especially difficult for him to find a girlfriend, or for a girl to be (and stay) his girlfriend.
To put it all in context:
"I WAS A LONELY TEENAGE BRONCIN' BUCK
WITH A PINK CARNATION AND A PICKUP TRUCK"
I was a moody, excitable, somewhat troubled teenage boy from a backwards rural area (the pickup truck reference) who was difficult to get along with, and who had no girlfriend because of it, even though I desperately wanted a girlfriend....
Hope this helps!