Alcohol can enter your system via the sweat and sebaceous glands in the scalp, or any dermal exposure for that matter. Listerine also contains about twice as much alcohol, by volume, as wine.
However, let's assume that the exposure did result in EtG synthesis even though it probably didn't. 1) The amount of EtG created would be very minimal and probably below the level of detection (LOD) even a few hours after use, and 2) even if it peaked above the LOD that would have occurred about 4-5 hours after use and then dropped below LOD within a couple of hours.
EtG does not remain in your system. It follows a standard elimination curve (see USDTL below).
There are no studies involving the use of Listerine on hair although many hair care products do contain significant amounts of ethanol. Regular use of these products have resulted in positive Hair Alcohol Tests (HAT) but even in those cases there was no record of a concurrent positive UA.
So I feel confident in stating that given the time span involved you will not test positive by UA. USDTL Monograph NOTICE:
Any person or entity that is subjected to or relies on EtG testing should read this Advisory issued by the US Department of Health and Human Services, September 2006 and remains in effect as of date of this posting. Copy available here > US DHHS Advisory