Aside from issues that may be caused by egui itself.
Possible causes in GodotEgui:
Check your console log for the following error "WARNING: _prefill_polygon: poly has too many indices to draw, increase batch buffer size. At: ./drivers/gles_common/rasterizer_canvas_batcher.h:1628"
If this is the case, in your Godot's "Project Settings" go to "Rendering -> Batching" and set the batch buffer size to 65535. While it is possible to be lower, egui expects to be able to use at least a 16bit indice value for the GUI.
egui is designed with the assumption that each egui::Context behaves as an explicity "application" so when using multiple GodotEgui nodes it is possible for the overlap between the nodes to occlude nodes on a lower Z Level.
If neither of the bugs above occur and your UI renders correctly when using eframe or other native egui solution, this may be an issue with GodotEgui please feel free to open an issue with a link to a project that can reproduce the issue.