Thanks for the replies Dan and Jayanta.
First, while getting some sample of the point data I realized that they gave me the wrong values in the wrong field. Latitude degrees in the longitude field. After correcting the data, the points show up onto my coastline data that was in the St.Kitts and Nevis projection.
Dan: The -62.5 central meridian does make sense as the island is very close to that longitude. Now that my data lines up, is it worth changing the shapefile data's projection to something more mainstream? Would it be as simple as putting the central meridian back to 0? I will be the only one using the data and only short term, so I'm not sure.
I have attached the coastline layer which has the modified projection. No sample points as I found that issue.