“Balloon juice” in diner lingo means seltzer or carbonated soda water. The carbonation fills one up, like a blown balloon. The term “balloon juice” has been cited in print since at least 1973.
 
 
Wikipedia: Diner lingo
Diner lingo< is a kind of American verbal slang used by cooks and chefs in diners and diner-style restaurants, and by the waitresses to communicate their orders to the cooks. It is virtually unknown outside the US.

 
History
The origin of the lingo is unknown, but there is evidence suggesting it may have been used by waiters as early as the 1870s and 1880s. Many of the terms used are lighthearted and tongue-in-cheek and some are a bit racy or ribald, but are helpful mnemonic devices for short-order cooks and staff.
 
List of terms
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Balloon juice/Belch water/Alka Seltzer: seltzer, soda water
     
Google News Archive
24 June 1973, Sydney Morning Sun-Herald (Australia), “A book of words mother didn’t like” by Nigel Palethorpe, pg. 91, col. 2:
Balloon juice, used to mean soda water (it blows you up); and a balloon-juicer or bun-strangler was a teetotaller.
(Eric Partridge’s Dictionary of Slang—ed.)
 
27 June 1991, Los Angeles (CA) Times, “Shake One in the Hay” by Charles Perry, Food, pg. H2:
Balloon juice: seltzer.
 
Google Books
How to Talk American:
A guide to our native tongues
By Jim Crotty
Boston, MA: Houghton Mifflin Company
1997
Pg. 79:
balloon juice: champagne or seltzer.
   
Google Books
Cassell’s Dictionary of Slang
By Jonathon Green
Cassell
2005
Pg. 61:
balloon juice n.’ [late 19C+] 1 soda water; thus balloon juke lowerer, a teetotaller, who only drinks or “lowers” soda water.
(...)
[? gaseous nature of soda-water; contemp. use is WI only]