They can indeed! Eevee will generally have different ways of communicating their desires. Some take on more of an interest in a certain element (for example, getting excited upon seeing an electric-type battle or seeming interested in learning moves of a specific type). Others may take on habits that lean more towards a certain element or habits that are shared by certain evolutions (e.g., becoming more active at night or showing more affection towards you). And still others may make things particularly easy by showing a keen interest in a specific evolution whenever they appear in media or are otherwise in the vicinity.
In other words, figuring out what your eevee wishes to evolve into is actually the easy part. It simply requires you to pay close attention to what seems to catch your eevee’s attention the most. When you’re almost certain you know what your eevee wishes to evolve into, show it a picture of the evolution in question and explain what that pokémon is and what it can do. If your eevee still seems interested after that, then you’ll know for certain.
After that, it’s just a matter of obtaining the right conditions for it to evolve, and that is the hard part.
The Pokédex According to Bill 