"Can they" versus "will they" are very different questions.
Companies can take action if you make use of just about any recognizable element of their stuff. Even if you don't use it exactly, if it is recognizable that is enough to sue.
Generally -- BUT NOT UNIVERSALLY -- companies will officially ignore or 'not notice' minor elements. Brands survive based on their popularity, and fan creations can help spread their brands.
When it comes to major elements companies take action. If you prominently use major characters or major elements or major themes they'll immediately take you down. If your product touches on story lines the company wants to use (even if they don't use it yet) they will take action. Products that started small as fan creations but grow into a large enterprise are likely to be shut down. Even if they quietly ignored the product when it was small, when it grows large enough it starts to take sales, starts to compete, starts to influence their brands, and they won't like that.
Can EA get offended by the fact that in my game a bad person plays their game?
Yes they can. Exactly how offended and your risk of a lawsuit depend on all the details.
Can EA sue me for using their game boot screen, and will it be justified?
Yes they can, and they would probably be justified.
Note that they usually don't start with a lawsuit. The first shots fired are usually a Cease and Desist letter.
A much more difficult question is if they will. The easiest way to avoid it is to contact them up front, describe the details of your plans and negotiate to pay a pile of cash in exchange for permission. This is easiest to do if "you" are actually a corporation with an established name. Otherwise, the second best way to avoid it is to be really small and never succeed commercially.
