I am wondering if I am totally dating myself with short rounded square nails.

I have preferred this shape since the 90s. I love the look of other shapes on other people (I enjoy so many of your lovely posts here!), but I dislike the feel of longer nails on myself, and any shape other than square or squoval looks funny to me on my own hands, even though it can emphasize short fingers.

Do squarish shapes scream "old lady" or does anything go these days as long as they are well maintained?

If color matters, I generally stick with neutral these days. I will go through phases where nails are more integral to self-expression, but currently I just aim for subtle and understated.