Can I replace a tiled shower with a fiberglass shower pan?
Short answer:  He/She may be correct but ask them what problems they forsee running into that made them decide that.

Long answer:
This should be possible but I'm not sure it will be visually appealing and you might have problems finding a premade pan that fits your space.  You may also need to redo the drain plumbing since fiberglass pans have standard locations for the drain unless specially ordered (usually directly in the center or centered along one of the short walls).  

You will also need to remove the bottom row of tile on the walls above the height that the pan will occupy and then retile that row to properly overlap the pan edge for drainage.  The question here is if you have sufficient space between the studs in the wall and the existing tile to overlap without there being a visible change in tile angle on the bottom row.

You might also have to remove some drywall (or whatever you have covering your studs adjacent to the shower bay) in order to get the pan to slide in and then re-patch that area when finished.