I would say no, you don't need any special tilemap programs.
Personally, I find tilemap software to be too limiting and just use Photoshop, which is something I'm more comfortable with. You could achieve very similar results with other programs, such as GIMP or Paint.NET, both of which are free.
It might be easier to make one base tile (say 32x32px) which you would duplicate and use as starting point for the rest of your tiles. It's also best to draw all your tiles in the same sheet, instead of as separate images so you can compare as you go, but if they all use the same base tile this shouldn't be a problem.
Most paint programs I know can set grids so you don't have to worry about misalignment, which is pretty much what tilemap programs do anyway.
I hope that helps.