Trying to add uv project modifier on eye but eye gets stretched up and down Eye before applying uv project modifier

Eye after applying uv modifier to empty bone.
As you can see, I don't know why but eyes are stretched out from top to bottom...

And when I move the empty slightly to the left, it goes like this...

UV setting is set to like this.
Why is this happening? I'm so confused... Please help me.
Thanks in advance.
The reason the texture isn't projecting right may be due to the lack of Coordinate and Mapping nodes, like so:

With the Texture Coordinate set to UV, the UV Project and UV Warp modifiers should start functioning properly. You can quickly add these nodes in with the built-in Node Wrangler add-on, using Ctrl+T after selecting the image texture node.
Note that in this example, I have the X and Y scale on the Mapping node set to -1, in order to flip the centered texture along those axis. Those properties can be animated as well, but the eye rig should be able to take over from there.