I'm looking for someone to sell me cardboard cartons just like juice
or milk cartons but with the following specifications: 2.25" square
and 6.5" high with the seam on the outside.  The polyethylene coating
is essential.  I don't want a gable or any graphics, and the carton
should have one open end.

Have you tried searching in the Thomas Register? 
http://www.thomasregister.com/
A search in product and polyethylene brought up a list
of cardboard manufacturers. 
Check your local library for manufacturer directories
for your state.

Hello -

If you're still interested in an answer, I have found a company that
can produce what you're looking for in various quantities (low to
high)... however, they cannot "seal" the bottom of the carton.  Is
liquid actually going into these cartons?  Since you've expired this
question, you'd have to repost a question to my attention if you'd
like this as an answer.  Thanks.

jbf777