The insurance company made one last push to settle but the family dug their heels in. They wanted their day in court.
what does "made one last push to settle" The company were making a final attempt to settle the case or, put another way, the company made a final offer. Another phrase you might see in such circumstances would be "made a lst ditch attempt to settle". In legal terms this is a common situation where the insurance company, or one party to a court action will try to bring the case to settlement right up to the last moment before the court case begins. At the last hour the "have a final roll of the dice to settle".
"dug their heels in" - this is describing the reaction, it expresses that the family would not change their minds, they would not be swayed in their resolve, they would not be persuaded. Another phrase you might see in such a situation would be "stuck to their guns" or "would not be moved" or "held firm".
"want their day in court" mean? This is an expression used where one party to a court action wants the opportunity to have their views heard in court. It is a phrase that can often be heard where people believe that the party who "wants their day in court" could have settled the case by accepting the offer but refused the offer because they wanted the opportunity to have their position made public by giving evidence in court or, by being able to challenge the evidence of the other party in court.
Hope this is helpful for you :)