The only uses of 'neck' as a verb I am aware of are 1. to kiss (quite old-fashioned). This doesn't make sense in this context 2. to drink quickly, e.g. to down a shot in one. It's possible it is therefore being used to mean eat.

A Texas trinity isn't a drink, it's a dish of barbequed brisket, ribs, and sausage. I've never heard the word "necked" used in relation to it. I would consider this a non-standard usage.