Style is determined more by the game than the coach. This England team played defensively because they have no true attacking midfielders. And by that I mean a player who usually plays deeper but orchestrates the midfield (Pirlo, Fabregas, Xavi, Mikel Arteta). The closest England have is Rooney dropping into midfield to collect the ball. Instead England have defensive midfielders (Parker, Gerrard, Milner) or they have wingers (Walcott, Young, Adam Johnson). There isnt even anyone like Nasri, or Ozil, an attacking midfielder who really likes to have and keep possession of the ball.
My point is, England don't have the players to keep the ball. Its not Hodgson, its just the way football goes. Maybe Wilshere could change that or OX. I'm not as familar with Mchearan, Powell or Barkley, and my impression of Rodwell is as more like Gerrard, a tackler, shooter, running type of midfilder rather than an orchestrator.
England has immense quality in the boxes. They defended expertly and put in a number of good crosses. Carroll played well also with his limited minutes. He is a much better forward for England than Welbeck. Caroll can win the clearances from a primarily defensive set-up where as Welbeck can only flick the ball on and hope for good outcomes. Welbeck would have been better in a team that could find his feet often and close to goal. For example the combination with Rooney at the top of Italy's box when Welbeck placed the shot well wide. Everything leading to the chance was well suited to Welbeck's style.
To directly answer your question, Hodgson didn't tell the England players to give the ball away and stick to defending. He didn't tell them to camp in there own half and attack on the counter. It's simply that England currently doesn't have the quality to keep the ball in the middle. All Roy did was make the decision to put Milner instead of Walcott. Essentially acknowledging the fact that, "Yea England is gonna have to play on the back foot."