The battle between the Blefuscu and the Lilliputians began several years before Gulliver met them. The problem was that the emperor of Blefuscu (who is ancestor of the current one), cut his finger when he cracked open an egg on the wider side. For this reason, he passed a law to be followed: That all Lilliputians have to break their eggs on the smaller part of the egg instead of the wider part like what they do.
Reldresal explained to Gulliver that, as a result of this command there were battles among the group, books were written about it, and it was basically a really big deal. Those who rebelled against the egg law where accused of violating religion, or the Brundrecral, like they named it. The Blefuscu said that if they were real believers they would do as they were told and break the eggs the way that the emperor said to break them.
So, basically the Lillputians were accused of refusing to follow the religious imposition of the emperor by not willing to break the eggs the way they were told to.