In the opening scenes, Billy Bob Thornton is writing his goodby notes on a, what seems to be, a pizza box with a Parker Jotter pen.
It is a red plastic barrel Jotter pen. It is not very clear but I think it is the translucent version.
In a very messy kitchen between empty bottles and a lot of trash
Billy Bob Thornton is using the Parker Jotter to note down his last wish
Of course, he doesn't go through with his plans, as it would be a bit depressing for a Christmas movie and also very short.