The ousted lawmakers, Reps. Cameron Henry and Joe Harrison, said they believe they were replaced because they questioned the Jindal administration and disagreed with his budget decisions.
Their two replacements voted Friday for the outsourcing plan.
Nichols said the contract with Blue Cross will save $20 million annually, with about $9 million of that for state agencies and the rest split among local school boards and state employees. The Legislative Fiscal Office instead pegged the savings at anywhere from $11 million to $18 million.
"We think there is going to be cost savings. We just cannot predict what that cost savings is going to be," said Travis McIlwain, who analyzed the contract for the Legislative Fiscal Office.
Rep. Katrina Jackson, D-Monroe, a vocal critic of the outsourcing plan, said additional budget documents only raised further questions about the Blue Cross contract. She questioned whether it will cost the state more money.
The governor hadn't intended to seek budget committee approval, but the attorney general's office said the contract required it.
When she asked to delay the legislative decision last week, Nichols said she wanted more time to address lawmakers' questions about savings and other aspects of the contract. Over the last week, the Jindal administration sent letters to public school boards about the money they could save with the change and tweaked the Blue Cross contract to add more legislative oversight.
"We did take your feedback very seriously," Nichols told lawmakers.