Foreign players need to meet government work permit criteria regarding caps, appearances in competitive international games and the like. EU players, like any EU worker, can just rock up.
That is a completely separate issue to foreign/HG squad designations.
And I'm pretty sure every single one of our foreign EU players would meet the criteria for a work permit today.
Not to mention the FA & home office would just loosen their work permit regulations because the fact is they want footballers paying millions in tax in the country, and the only reason the current rules are so strict on non-eu players is because of the fact they can't regulate EU nationals from coming in.