I'm trying to make an L shape where the gradient would go from the edge of the path to the center. In the image currently, I have 5 squares (ignoring the white squares) with a reflected gradient. The issue in this image is the corner. I'm not sure how I can seamlessly connect it to its neighbor. Any ideas?
If you draw your L as a single shape or stroked path, not separate pieces, then you could use a Stroke layer style using a gradient,set the position to Inside, and the style set to Shape Burst, and make the size big enough to cover the hole in the middle.
If you don't want the gradient to extend to the ends of the L, you could cover over the ends with black rectangles on a new layer, or white squares if you prefer.