European cultures had crossed buns representing the division the four corners of the earth ( not the Christian cross) long before Christianity adopted them for their own celebrations, same with the worship of the Egg (Easter) which was to usher in the Spring moon (Northern Hemisphere Only) which signified fertility and time to have their crops planted for best results.
So they painted eggs, as the Egg symbolized fertility which was linked to the moon.
So both would likely still be used in such celebrations today.
It's funny how those ancient growers worked out that sowing their crops at that time was so valuable, as it works.
Ancient traditions do hold some reliable knowledge!
Easter has become a symbol, similar to Christmas that it's about god when in reality it really wasn't.
Sure, Easter is the day that god got resurrected but it doesn't mean eggs relate to him more then people want them to, which is normally that the eggs relate to Jesus Christs tomb, which is true.. To an extent.
In actual fact, the first eggs to be used in the "Religious" fashion of tombs predated Christianity by 1,000s of years, the earliest known ritual of this kind was when the Egyptians and the Sumerians commonly placed decorated ostrich eggs in the tombs of their dead, there are lots of other reasons why we have Easter as we know it today, such as Lent and Fasting in Western (American) Christians.
Would they exist without Christianity? Yes, probably not in the sense that we eat them on a glorified day of the year, but they would still "exist" in these predated tombs.
The history of the dyed or painted eggshell goes way back. They are decorated ostrich eggs that date back 60,000 years ago in Africa. Decorated ostrich eggs, and representations of ostrich eggs in gold and silver, were commonly placed in graves of the ancient Sumerians and Egyptians as early as 5,000 years ago. Like many things, the Christian faith turned local customs into their own religious meaning to encourage people to join.
The ancient Greeks may have marked cakes with a cross. Some have claimed a connection with the goddess Eostre, but there is no historical evidence to support this; Bede, the sole source to mention Eostre, says nothing about her attributes or ceremonies, but the bible never mentions hot crossed buns either yet it is obviously part of the celebration. The symbol of the cross has been around since before the time too and is found in ancient sites around the world that had no relationship with the biblical god much less Christianity. Seeing that they decorated other things with it, it is likely they would have also decorated food with it.
Granted, they probably would not be named as such but they would still exist by some name.
Many (incorrectly) claim that Easter comes from Ishtar, the ancient Babylonian goddess of fertility, war, and sex (the last two are my favorite). This is inaccurate, and unfortunately the misinformation is still spread around like wildfire.
Some claim that Easter originates from Ostern, (modern German), which came from Ēastre (old english), of which comes from the Germanic goddess Ēostre.
While this may be true, it's no secret that many Christian holidays were initially pagan holidays that had been eaten up by the young Christian religion as it spread. This still doesn't quite answer the question of if these two items, Easter Eggs and Hot Cross Buns, would exist if not for Christianity.
If in some hypothetical world Christianity never made it out of the sands of the middle east, I believe the Easter, (in some different form) would still exist.
The eggs would still represent fertility and spring, (as the do now).
The predominately Pagan Easter bunny would still be around.
As for Hot Cross Buns... The origin of these is mysterious, but there's little risk in assuming that they come from more recent Christian tradition. If, by some chance, the true origin was lost with time, they might not have existed, (or perhaps in their current form), but as far as we can tell, this was a creation of the Easter holiday AFTER it had become Christianized.
A very interesting question, none the less.
Christianity has caused Christmas, Easter, Good Friday, (indirectly) Black Friday, Passover, and more. Without Christianity, we would not have any of these holidays. Easter eggs are a Easter tradition, so without Easter, we would not have Easter eggs. It's that simple and I need more words. There you have it.
Easter eggs symbolise Jesus empty tomb which is why. That is why it is called Easter because it is representing this religious holiday.
Hot Cross Buns are also designed because of Christianity with their cross on it.
They were both invented because of Christianity and to celebrate a Jesus life, without it may not exist.