From the links on the page you referenced.
The Franco-German initiative quickly aroused the interest of other countries. This was particularly true for Belgium that wanted to integrate its forces into an army corps. It was looking for a way to demonstrate its willingness to contribute to the construction of a European defence and security identity, while at the same time preserving its role within NATO. Since the Eurocorps offered such a solution, Belgium joined on June 25th, 1993.

The first Commanding General of the Eurocorps, Generalleutnant Helmut Willmann, assumed his appointment on October 1, 1993. The inaugural ceremony was held in Strasbourg on November 5, 1993 and was co-chaired by the defence ministers of the first three participating countries (France, Germany and Belgium).

Spain joined the Eurocorps on July 1, 1994, becoming the fourth member.

A symbolic moment for Eurocorps and European history was the participation of the Eurocorps in the French National Day military parade on 14th July 1994 when Eurocorps soldiers marched down the Champs Elys\ufffdes in the centre of Paris. One week later the Eurocorps paraded in Brussels for the Belgian National Day.

Luxembourg joined the Eurocorps on May 7, 1996, as the fifth Member State.
Hmmm, so individual nations in the EU are not members of the Eurocorps?

That sounds strange. After all, you claim it is the military branch of the EU.

Well, FIVE nations joined it.

In the meantime important decisions had been taken on the European Security and Defence Policy. These decisions have affected the role and structure of the Eurocorps. On May 29, 1999, at the French-German Summit in Toulouse, it was proposed to place the Eurocorps at the disposal of the European Union for crisis response operations.
Hold it. If the Eurocorps is the military branch of the EU (since 1992, so you claim), then WHY, in 1999, did there have to be a specific proposal to place the Eurocorps at the disposal of the EU?

This was accepted by the other member states and formally announced at the European Union (EU) Summit in Cologne (3-4 June 1999).
Okay, so this is from the EU nations.

Around the same time, at a meeting in Luxembourg, Eurocorps member states decided to transform the Eurocorps into a Rapid Reaction Corps available to both the European Union and the Atlantic Alliance.
And THIS is the Eurocorps nations.

But, according to boxley, THEY ARE THE SAME NATIONS.

But, according to that website, they are NOT the same nations.

Boxley says they are the same nations.
The website says they are NOT the same nations.