Don't duct tape the lid shut. They will almost certainly cut it open to search it then. That's a red flag. If you lock luggage, then the same thing happens. They'll cut the lock and search.
Instead, buy a couple of luggage straps & strap the cooler closed.
This works well. I've brought quite a few glass bottles & jars of food items home from other countries before. I just use a hard sided cooler on wheels. This way, they are less likely to throw it around. You want it heavy enough that it's awkward to lift very far. They'll stack other luggage on top of it, but that's okay if you use a STURDY cooler.
I have yet to have a jar or bottle broken using this method.
For eggs, I would use standard styrofoam egg cartons (they seem more secure) and then lay a piece of bubble wrap inside the lid so that it will be a "tighter" fit. Wrap each in a piece of bubblewrap. Stack neatly in the cooler. Put clothes in ziploc bags (gallon sized?) and put them around the eggs to stabilize and fill empty space. You want to ease the search job so they are less likely to get broken.
Use magic marker to write your name and address directly on the outside AND inside of the cooler. Lay a sheet of paper on top inside the cooler stating the contents of the cooler. They may spend less time searching (or may not...)
I would even risk carrying hatching eggs this way.
