Can I really catch a nail disease from the shower at my gym?

Yes. Yes it is possible to pick up a fungal infection from an unclean shower at the gym, however, is is probably more common to pick up athletes foot (tinea pedis) before you would pick up a toenail fungus.
Yes. If the previous person had a fungal disease and/or the shower has not been cleaned/disinfected recently, you can catch the disease.