I came across this issue while idly checking to see if any of the Eurosceptics had had the honesty to admit that the death penalty has not in fact been introduced as a result of the Lisbon Treaty. (I think you can guess the answer.) In fairness, the more mainstream swivel-eyed loons haven't touched this one, apart from Ashley Mote MEP noting that the EGF website talks of the EU constitution, thus a "clear admission" of what the Lisbon Treaty really is (because obviously it is more likely that the governments of 27 countries would collectively lie to their people than that a poorly-written website might be inaccurate).
It is paranoid fantasy of course. The EGF, whose total staff numbers 30 rather than 3000, isn't even an EU body; it is a pooling of expertise on what one might call robust policing methods by France, Italy, the Netherlands, Portugal, Spain and Romania, with input also from Poland and Turkey. It is not deployable within the EU - this is not made terribly clear on the site (and is a point missed even by some people who should know better) but it is explicit in reference to the Santa Maria de Feira and Nice European Council conclusions, and to the Petersberg declaration, all of which is EU-speak for deployment to external crises in support of (or substitution for) the UN or similar bodies.
My own first encounter with it was back in 2005 (see recommendation #2 of this report, which was not in fact adopted by the powers that be). Looking at the website now, what strikes me is how little the EGF appears to have actually done in its five years of existence. Members of its secretariat get deployed hither and yon (this year so far to Afghanistan, Bangladesh and Haiti) and as a body it has taken responsibility for training Afghan police in Mazar-i-Sharif and coordinating the police aspects of the EU military mission in Bosnia. But in order to be the effective bogeyman of Europhobe myth, it needs to raise its game considerably!