Many communities are living in fear of being kidnapped for ransom as the crisis spreads further south.
A 50-year-old Thai suspect was arrested with the eggs in a "homemade incubator" carried as hand luggage.
It is thought to be part of efforts to consolidate the former coup leader's control.
Eswatini is one of 12 countries, and the only one in Africa, that has diplomatic relations with Taiwan.
His brother tells the BBC his family have not been allowed to see him "in all these years".
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