While a macro lens is not necessary to capture a brown recluse spider on film, using one is likely to result in significantly clearer photographs. However, there are a number of other considerations, such as magnification and focal length, that will dictate the quality of the image.
Sometimes lenses will be mislabeled as "macro." It is important to look for those with a 1:1 or 1:2 magnification. This information will be in the lens manual, if not on the lens itself.
It is also important to make sure that the lens has a fixed focal length. Zoom lenses, while allowing for a closer image of brown recluse spiders, are not true macro lenses.