I had 1 beer with dinner at 6 pm I tested for a nursing board random test at 10 am the following day. Will the EtG show a positive for alcohol?

Yes..

Of course it will be detected, but at what level is unknown. It really depends on the what the acceptable levels for passing are, and the nursing professions have a quite low threshold if not a zero tolerance.

Uncertain if a fail triggers a retest, or further testing, or denial of a license. Depends on the state/school/hospital.

The drug test for nursing boards, at least in my state, tests for narcotics and illegal substances, not alcohol.