Hi,
This is most likely not a HikaShop issue but a Joomla menu state one. The symptom (an item becomes unpublished after a save and refuses to be published again) is the well-known Joomla cascade behaviour when the menu's invisible root item is itself unpublished, or when the menu's nested set (the lft/rgt columns of the #__menu table) is out of sync. Two things to try :
1. Open Menus > Menu Items, click the toolbar dropdown and hit "Rebuild". This recomputes the nested set and usually unblocks this kind of stuck-unpublished state.
2. If the rebuild does not help, the menu's root item is probably unpublished. The root is hidden from the menu list UI but exists in the #__menu table with type = "Menu_Item_Root" and a parent_id of 0 per menutype. You can flip its published column to 1 directly in the database (phpMyAdmin or similar).
After that, the published checkboxes on the menu items should hopefully start working again.
If neither step changes anything, I would recommend posting on the Joomla Forum.