House Republicans on Saturday called on the Senate to vote to delay ObamaCare's individual mandate since businesses will not be required to provide healthcare coverage until 2015.
Young and Griffin accused Democrats and the White House of ignoring the needs of individuals by opposing their legislation.
"Many Democrats voted to stand with big business and against fairness for individuals and families," said Young.
"President Obama threatened to veto our proposals altogether. We take that to mean he thinks it's fair to let businesses off the hook while leaving middle-class families in harm's way."
Griffin told the story of a "21-year old Hispanic-American who runs a small business in my district" who says he cannot hire more workers because "ObamaCare makes him choose between new, higher insurance premiums or hefty fines."
Republicans have stepped up their criticism of the healthcare law since July 2, when the Treasury Department announced the delay of the employer mandate.
The policy requires businesses of 50 or more workers to provide health insurance, and involves an elaborate reporting system.
Administration officials said their deferral responded to requests from business for more time to understand that system.
Republicans quickly turned the argument on the White House, saying individuals deserve as much flexibility as companies.
The Senate is unlikely to vote to delay the individual mandate because that move would cause a spike in insurance premiums on the individual and small-group markets, dooming ObamaCare's new insurance exchanges.