In act IV.iv what does he mean by this "To all that fortune, death, and danger dare, Even for an eggshell..."
Hamlet is commenting on the fact that the Norse and Polack armies are about to fight over a worthless piece of land. Thousands are likely to die in this battle, and the land being fought for isn't even large enough to bury the dead. But both sides are willing to die. He realizes that the land is not what the two sides are fighting for, but rather that "honor's at the stake." After hearing of this imminent battle, Hamlet realizes that if anyone has a cause worth dying for, it is he, considering the fact that his father has been murdered, his mother stained and his uncle has pretty much stolen the crown that should have gone to Hamlet. He vows that from that point forward his thoughts will "be bloody, or be nothing worth," meaning that from that point on his only thoughts will be on revenge against Claudius.
Enotes has great side by side translations of Shakespeare's plays. Here is the modern version of the quotation about which you asked:
"Examples, as obvious as dirt, urge me.
Look at this army, of such size and power,
Led by a delicate and tender prince,
Whose spirit, with divine ambition puffed up,
Makes faces at the invisible event,
Exposing what is mortal and unsure
To all that luck, death, and danger dare to do,
Even for an eggshell."
For a full translation click the link below.