There were also civilian casualties after some of the recruits' friends and family used a bus to block the road to help them escape, according to the unconfirmed reports.
Conscription in Eritrea is compulsory.
The Eritrean authorities have not commented on the alleged incident.
Rights groups consider Eritrea to be one of the world's most repressive states.
In 2015, it ranked bottom of the World Press Freedom Index, published by media watchdog Reporters Without Borders (RSF).
Conscription in Eritrea can last for decades and is one of the main reasons tens of thousands flee the country every year.