Afeworki spurned a call by Yemeni President Ali Abdullah Saleh that both leaders settle their differences "through dialogue and understanding" to avoid an escalation of violence "harmful to stability, security and peace in the region."
In a telephone call with Saleh Afeworki said Eritrea would "by no means engage" in any diplomatic activity with its neighbor, AFP reported.
"It has no desire whatsoever to be dragged into the current diplomatic and media tit-for-tat designed to further aggravate the fabricated problem," the statement said.
Saleh had also spoken to Djibouti President Ismail Omar Guelleh on Monday.
The Djibouti foreign ministry issued a statement after the frontier clashes accusing Eritrea of seeking to destabilize the Horn of Africa region by attacking Djibouti and vowed to use all available means to defend itself.
Following the shootings some 100 Djibouti-based French soldiers were deployed along the border with Eritrea, French army headquarters said Saturday.
"We have set up an advanced logistical base in the north of Djibouti that is removed from the combat zone," said French navy commander Christophe Prazuck.
France has 2,850 troops stationed in its former colony.
The United States has more than 1,200 troops stationed in Djibouti, which hosts an anti-terrorism task force for the Horn of Africa.
Hundreds of Eritrean asylum-seekers have fled the country, some reaching Egypt via its southern border with Sudan either hoping to receive permanent refugee status from the United Nations agency for refugees UNHCR, or to make it to neighboring Israel.
But Egyptian authorities have admitted to deporting around 200 asylum seekers back to Eritrea and are "preparing to deport a large number of Eritreans back to their country," moves criticized by human rights group Amnesty International.
"The Egyptian authorities forcibly returned a group of around 200 asylum-seekers to Eritrea on the night of 11 June, and are preparing to forcibly return a further 1,400," Amnesty International said in a statement.
"Most asylum-seekers returned to Eritrea are likely to be arbitrarily detained incommunicado in inhumane conditions from weeks to years," Amnesty said.
"They will be at serious risk of torture or other ill-treatment, particularly those who have fled from compulsory military service."
The Egyptian foreign ministry Sunday said it will allow United Nations officials access to Eritrean asylum seekers on its territory, according to AFP.
The latest frontier incident was the first since a tense standoff on April 16 when Djibouti accused Eritrean forces of digging trenches on both sides of the border, infringing several hundred meters into Djibouti territory.
The two nations have clashed at least twice over the border area at the southern end of the Red Sea.
In April 1996 they almost went to war after a Djibouti official accused Eritrea of shelling Ras Doumeira.
In 1999, Eritrea accused Djibouti of siding with Asmara's arch-foe Ethiopia, while Djibouti accused its neighbor of supporting Djiboutian rebels and having designs on the Ras Doumeira region. Eritrea has denied this.

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